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

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

Proof. Use normal graphs \(\Sigma(v)\) over \(\Sigma\) and pull their expansion back to \(\Sigma\). The map \[\Phi:C^{2,\alpha}(\Sigma)\longrightarrow C^{0,\alpha}(\Sigma), \qquad \Phi(v)=\Theta_Q(\Sigma(v))-c\] is smooth near zero. Let \(\nu\) be the outward normal and \(\mathrm{II}\) its second fundamental form. The normal-variation formulas give the linearization \(L=D\Phi(0)\) explicitly as \[Lv=-\Delta_\Sigma v+2Q(\nu,\mathop{\mathrm{grad}}_\Sigma v) +\bigl(\mathop{\mathrm{tr}}_\Sigma\nabla_\nu Q-|\mathrm{II}|^2 -\mathop{\mathrm{Ric}}(\nu,\nu)\bigr)v.\] The hypersurface dimension is \(d-1\), so \(\Theta_{Q-cg/(d-1)}=\Theta_Q-c\). This constant shift has zero derivative under hypersurface variations; hence \(L\) is also the MOTS stability operator for \(Q-cg/(d-1)\), with principal part \(-\Delta_\Sigma\). Stability gives a principal eigenvalue \(\lambda\geq0\) and a positive smooth eigenfunction \(\varphi\) (Andersson et al. 2008, Proposition 5.1). If \(\lambda>0\), take \(v=s\varphi\). Smooth dependence gives \(\Phi(s\varphi)=s\lambda\varphi+O(s^2)\) in \(C^0\), which is positive for all sufficiently small \(s>0\). The graph velocity is positive as well.

Suppose \(\lambda=0\). The kernel of \(L\) is spanned by \(\varphi\), and the adjoint kernel is spanned by a positive smooth function \(\varphi^*\). These are the principal-eigenfunction facts for a scalar elliptic operator on a compact connected closed manifold (Andersson et al. 2008, sec. 4, Lemma 4.1). For example, simplicity of the kernel also follows by writing a kernel element as \(v=\varphi w\) and applying the strong maximum principle to the resulting equation for \(w\), whose zeroth-order term vanishes. Consider the augmented map \[\mathcal G(v,a) =\left(\Phi(v)-a,\ \int_\Sigma v\,\,\mathrm dA\right).\] Its derivative at \((0,0)\) is \[ (v,a)\longmapsto \left(Lv-a,\ \int_\Sigma v\,\,\mathrm dA\right). \tag{21}\] This map is an isomorphism. Indeed, to solve \(Lv-a=f\) with \(\int v=b\), the Fredholm compatibility condition uniquely fixes \[a=-\frac{\int_\Sigma\varphi^*f\,\,\mathrm dA} {\int_\Sigma\varphi^*\,\,\mathrm dA}.\] The equation for \(v\) then has a solution; adding a multiple of \(\varphi\) imposes its prescribed integral uniquely. Elliptic estimates and this one-dimensional normalization give a bounded inverse between the indicated Hölder spaces.

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

In either case positive graph velocity and compactness give the asserted bounds on \(|\nabla s|\). Smooth elliptic bootstrapping of the graph equation gives smooth leaves in the zero-eigenvalue case. ◻

Proof. If \(D=\varnothing\), use \(\operatorname{dist}(y,D)=+\infty\); then \(D_z=\varnothing\) and the assertion is vacuous. Otherwise, take a support \(\phi\) to \(D_z\) at \(x'\). Since \(x'\in\partial D_z\), its distance from \(D\) is \(z\). For sufficiently small radii the relevant distance neighborhood of \(\partial D\) lies in a fixed compact interior neighborhood. There is therefore a nearest point \(x\in D\) and a length-\(z\) minimizing unit-speed geodesic \(\gamma:[0,z]\to N\) from \(x\) to \(x'\). This nearest point lies on \(\partial D\): if it were interior to \(D\), moving a short distance along \(\gamma\) toward \(x'\) would decrease the distance. The entire segment lies in the fixed smooth compact neighborhood and is shorter than its injectivity radius. The ball \(\overline B_z(x)\) is contained in \(D_z\) and touches its exterior support at \(x'\). Its tangent plane there therefore agrees with that of the support and \[ \frac{\nabla\phi(x')}{|\nabla\phi(x')|}=\dot\gamma(z). \tag{24}\] Let \(P_t:T_xN\to T_{\gamma(t)}N\) denote parallel transport and put \(P=P_z\).

For every \(X\in T_xN\) solve the Jacobi boundary problem \[ J_X''+\operatorname{Rm}(J_X,\dot\gamma)\dot\gamma=0, \qquad J_X(0)=X,\qquad J_X(z)=PX. \tag{25}\] Here primes denote covariant differentiation along \(\gamma\). The short interval and absence of conjugate points make this problem uniquely solvable. Its estimates can be seen without division by an uncontrolled quantity. In a parallel frame write \(j_X(t)=P_t^{-1}J_X(t)\) and \(\mathcal R(t)\) for the curvature coefficient. The integral equation is \[j_X(t)=X+tA_X- \int_0^t(t-s)\mathcal R(s)j_X(s)\,\,\mathrm ds, \qquad A_X=\frac1z\int_0^z(z-s)\mathcal R(s)j_X(s)\,\,\mathrm ds.\] It implies, by absorption for small \(z\), \[ \begin{aligned} \sup_{0\leq t\leq z}|j_X(t)|&\leq2|X|, \qquad |A_X|\leq Cz|X|,\\ \sup_{0\leq t\leq z}|j_X(t)-X|&\leq Cz^2|X|. \end{aligned} \tag{26}\] The second line follows from the same integral equation and the first-line bounds. Also \(\langle J_X,\dot\gamma\rangle\) is affine in \(t\) and has equal endpoint values. Thus \(\langle A_X,\dot\gamma(0)\rangle=0\).

We may therefore prescribe the first jet of a smooth unit vector field \(V\) near \(x\) by \[V(x)=\dot\gamma(0),\qquad \nabla_XV(x)=A_X.\] To realize it with uniform second-jet bounds, choose a smooth local orthonormal frame obtained by radial parallel transport from \(x\). In that frame take the affine coefficient vector with the prescribed value and first derivatives, and normalize its length. The orthogonality just proved means normalization preserves its prescribed first derivatives. The frame has uniformly bounded derivatives on a fixed small coordinate ball, and Equation (26) therefore gives uniformly bounded second derivatives of \(V\). This construction uses the radial direction and the ambient geometry, with no derivatives of the support \(\phi\).

Define the local endpoint map \[F_z(y)=\exp_y\bigl(zV(y)\bigr).\] Its differential at \(x\) is the terminal value of the Jacobi field with initial values \(X,\nabla_XV(x)\). By Equation (25), it has the exact property \[ F_z(x)=x',\qquad (\,\mathrm dF_z)_x=P, \qquad |(\nabla\,\mathrm dF_z)_x|\leq Cz. \tag{27}\] The last bound follows by differentiating the smooth map \((y,v,z)\mapsto\exp_y(zv)\) twice in \(y\), including the first and second derivatives of \(V\). At \(z=0\) the map is the identity and its covariant second derivative vanishes. Its \(z\)-derivative is uniformly bounded for the bounded jets just constructed, giving the displayed \(Cz\) bound. The exact middle identity, rather than a near-isometry estimate, is essential.

The map \(F_z\) takes \(D\) locally into \(D_z\), since its defining geodesic from any \(y\in D\) has length \(z\). Consequently \(\psi=\phi\circ F_z\) is an exterior support to \(D\) at \(x\). Its derivative is \(P^*\,\mathrm d\phi\), so its gradient length is exactly \(|\nabla\phi(x')|\) and, by Equation (24), its normal is \(\dot\gamma(0)\). The covariant chain rule gives \[\mathop{\mathrm{Hess}}\psi(X,Y) =\mathop{\mathrm{Hess}}\phi(PX,PY) +\,\mathrm d\phi\bigl((\nabla\,\mathrm dF_z)(X,Y)\bigr) \quad\hbox{at }x.\] Trace over an orthonormal basis of \(\dot\gamma(0)^\perp\) and divide by the common gradient length. The first term is exactly the corresponding normalized tangential trace at \(x'\), and the remaining term has absolute value at most \(Cz\). Finally, the tensor trace changes by at most \(Cz\), by the \(C^1\) bound on \(Q\) and parallel transport. It follows that \[\bigl|\mathcal E_Q[\psi](x) -\mathcal E_Q[\phi](x')\bigr|\leq Cz.\] The assumed support inequality at \(x\) proves Equation (23). There is no term containing \(z|\mathop{\mathrm{Hess}}\phi|\). ◻

Proof. All distance neighborhoods in this proof are relative to \(N\). Since \(D\) contains a collar of \(\Gamma_-\) and avoids \(\Gamma_+\), their new frontiers, at sufficiently small radii, lie in a compact subset of the interior. A minimizing segment ending on a new frontier has its nearest point on the interior frontier of \(D\); it cannot start on \(\Gamma_-\) because its covered collar has fixed positive width. Thus the proof of Lemma 9 applies unchanged. Choose \(\beta>0\) so small that its conclusion holds for \(0<z\leq\beta\), all these frontiers avoid the prescribed boundary, and \[ c+C\beta<c'. \tag{29}\]

We first construct a smooth inner enclosure without imposing any expansion bound on it. Smoothly approximate the continuous distance function \(d_D=\operatorname{dist}(\cdot,D)\) on \(N\) uniformly, with error less than \(\beta/32\), using a finite coordinate cover, mollification, and a partition of unity. Choose a regular value in \((\beta/3,\beta/2)\) of the resulting smooth function. Its sublevel set \(U\) has smooth interior boundary and, for example, satisfies \[ D_{\beta/4}\subset U\subset D_{3\beta/4}. \tag{30}\] The boundary \(\Gamma_-\) is contained in its relative interior and \(\Gamma_+\) is excluded. If a component of \(N\setminus\overline U\) meets no component of \(\Gamma_+\), fill it. This only deletes components of the smooth free boundary. After this filling the second inclusion in Equation (30) need not hold for its interior, but every remaining free boundary component still lies in the indicated thin neighborhood, and the first inclusion still holds.

Choose \(\eta\in C^\infty(N)\), \(0\leq\eta\leq1\), equal to one on that free boundary and supported in \(\{d_D<\beta\}\). Its existence uses the positive separation between this compact smooth boundary and the closed set \(\{d_D\geq\beta\}\); it requires no smoothness of \(d_D\). For a sufficiently large finite constant \(A\), the tensor \[ Q_A=Q-\tfrac12(c'+A\eta)g \tag{31}\] makes the free boundary of \(U\) a strictly negative inner barrier: \[\Theta_{Q_A}(\partial U) =\Theta_Q(\partial U)-c'-A<0.\] All prescribed outer faces remain strictly positive for \(Q_A\), because \(\eta=0\) there and their original expansion exceeds \(c'\). Apply the barrier part of Theorem 5 on each component of \(N\setminus\overline U\). Each meets \(\Gamma_+\) by the filling step and has nonempty inner boundary by connectedness of \(N\) and \(U\ne\varnothing\). We obtain smooth MOTSs \(\Sigma\) bounding a domain \(E\) which contains \(\overline U\) and avoids \(\Gamma_+\). They satisfy, with their normal out of \(E\), \[ \Theta_Q(\Sigma)=c'+A\eta\geq c'. \tag{32}\]

We claim \(\Sigma\) misses \(\{d_D<\beta\}\). Otherwise set \(z=\min_{\Sigma}d_D<\beta\). The first inclusion in Equation (30) gives \(z\geq\beta/4>0\). Moreover \(D_z\) lies on the inner side of \(\Sigma\). To see this directly, if \(d_D(y)<z\), a shortest path from \(y\) to \(D\) of length less than \(z\) cannot cross \(\Sigma\): such a crossing would have distance from \(D\) less than \(z\). Since \(D\subset E\), the whole path and \(y\) lie in \(E\). Closure gives \(D_z\subset\overline E\).

At a point attaining the minimum, a defining function for the smooth boundary \(\Sigma\) is therefore an exterior support to \(D_z\), with the same outward normal as in Equation (32). The parallel support bound and Equation (29) imply \[\Theta_Q(\Sigma)\leq c+Cz<c',\] contradicting Equation (32). This excludes the entire modification range, so \(\eta=0\) on \(\Sigma\) and proves Equation (28). The contained \(D_{\beta/4}\) gives strict enclosure of \(D\). If desired, fill any remaining complementary components which meet no designated outer face; this deletes smooth boundary components and preserves all stated properties. ◻

Proof. The large-sphere comparison above confines all negative-threshold regions to a fixed compact set, independently of how close the thresholds are to zero. Choose a slightly larger compact set \(\mathcal K\) containing this range and \(\mathop{\mathrm{supp}}K\) in its interior. Let \[m_0=\min_{\mathcal K}\mathfrak m>0, \qquad T_0=\sup_{\mathcal K}|\mathop{\mathrm{tr}}K|.\] First restrict both negative thresholds to an interval so close to zero that \[\max\{|c_b|,|c_w|\}\,T_0<m_0/8,\] while retaining Equation (14). Lemma 6 allows the required successive choices in that interval. Construct the two regions, select \(\Omega\), and fix their disjoint collars inside \(\mathcal K\) as above. All ensuing constants may depend on these fixed choices.

Here is an explicit offset construction. On each collar choose a smooth nonincreasing cutoff \(\chi(s)\), equal to one near \(s=0\) and zero near the collar’s far end. Choose \(\kappa_B>0\) so that \(1-\kappa_Bs>0\) throughout that collar. For a common amplitude \(a>0\), define the contribution of a black collar to be \(a\chi(s)(1-\kappa_Bs)\), and that of a white collar to be \(-a\chi(s)(1-\kappa_Bs)\). Extend by zero outside the disjoint collars and sum. This produces a smooth function on the manifold with boundary \(\Omega\), supported in \(\mathcal K\), with \(C_b=a\), \(C_w=-a\), and \(k_B=a\kappa_B\) on the respective collars. These remain valid bounds when a type is absent. Its \(C^1\) norm tends to zero with \(a\), even though the collar widths have already been fixed.

For \(h\in[b_-,b_+]\) and \(C\in[C_w,C_b]\), \[|h+C|\leq\max\{|c_b|,|c_w|\}+C_b-C_w.\] Choose \(a\) so small that \((C_b-C_w)T_0+\|\nabla C\|_\infty<m_0/8\). The first two terms of Equation (35) are then at most \(m_0/4\) on \(\mathcal K\) and vanish outside \(\mathcal K\). Finally take a fixed smooth positive weight equal to \(r^{-4}\) far out and multiply it by a sufficiently small positive constant. The end lower bound for \(\mathfrak m\) and \(\delta<1\), together with compact positivity, allow the choice \(\rho\leq\mathfrak m/4\) everywhere. This proves Equation (35). All remaining assertions were proved in constructing the regions and collars. ◻

Proof. Apply the scalar identity of (OpenAI 2026b, Proposition 4.1) in spatial dimension three, so its surface dimension is \(k=2\). To distinguish the two conventions temporarily, add a subscript \(\mathrm A\) to its variables. The exact dictionary is \[ \begin{gathered} t_{\mathrm A}=2t,\quad Z_{\mathrm A}=2Z,\quad l_{\mathrm A}(s)=l(s/2),\quad p_{\mathrm A}=p/2,\\ \mu_{\mathrm A}=8\pi\mu,\qquad J_{\mathrm A}=8\pi J, \qquad S_{\mathrm A}=S. \end{gathered} \tag{47}\] The quantities \(D,d,w,a,v,H^f,F,U\) are identical. In particular, \[\frac{2d}{2+p_{\mathrm A}v}=\frac{4d}{4+pv}=\chi, \quad u_{\mathrm A}=e^{t_{\mathrm A}}l\sqrt D=u, \quad 2\nabla Z_{\mathrm A}=4\nabla Z,\] and the companion metric \(e^{2t_{\mathrm A}}\bar g\) is precisely \(\hat g=e^{4t}\bar g\). Thus the divergence term, trace prescription and metric match without rescaling their flux. The physical density conversion gives \(8\pi(\mu+J(w))\); the ADM energy and momentum are not changed by that conversion.

For the quadratic expression, the companion coefficient \(s_p=p_{\mathrm A}+(k-1)/2\) is \((p+1)/2\), and \(dt_{\mathrm A}=2dt\). Its transverse square is \(|M+(p+1)a\nabla_\perp t|^2\). Combining its residual \(-[a^2|\nabla_\perp t|^2]\) with that square gives \((M+pa\nabla_\perp t)\cdot(M+(p+2)a\nabla_\perp t)\). The remaining gradient contribution is \(4(p+1)|dt|_{A_d}^2\). Its normal terms become \[(\chi-\chi^2/4)q^2+2(p+1)\chi qa\,\partial_e t -\chi Fq/2+3F^2/4.\] These substitutions give Equations (45) and (46). Smoothness at zero gradient is part of the invariant identity: the temporary axis has not been used to define its coefficients. The import is pointwise and requires neither an energy condition nor a solved divergence equation. ◻

Proof. With \(x=\partial_e t\) and \(y=\nabla_\perp t\), exact completion gives \[ \begin{split} \mathcal T={}&\tfrac12|T^{\operatorname{tf}}|^2 +|M+(p+1)ay|^2+[4(p+1)-v]|y|^2\\ &+4(p+1)d\left(x+\frac{\chi aq}{4d}\right)^2\\ &+\tfrac34\left(F-\frac{\chi q}{3}\right)^2 +\frac{4d(9-d)}{3(4+pv)^2}q^2. \end{split} \tag{50}\] Completing first in \(x\) leaves \(3\chi q^2/(4+pv)\); completing next in \(F\) leaves \(3\chi/(4+pv)-\chi^2/12=4d(9-d)/[3(4+pv)^2]\). The latter coefficient is at least \(32d/[3(4+N)^2]\). It controls \(dq^2\) polynomially. The \(F\)-square then controls \(F^2\), since \(\chi^2q^2\le dq^2\). The transverse gradient coefficient is at least three, and the shifted \(M\)-square controls \(M\) with an additional factor \((1+N)^2\). The squared \(d\)-norm of the shift in the \(x\)-square is \(\chi^2vq^2/(16d)\le dq^2/16\), so it also controls \(dx^2\). Finally \(|T|^2=|T^{\operatorname{tf}}|^2+(F-\chi q)^2/2\). This proves (48). If \(\,\mathrm dt=0\), the sole cross term is \(-\chi Fq/2\). Using \(|\chi Fq|/2\le\chi q^2/4+\chi F^2/4\) and \(0<\chi\le1\) proves (49). ◻

Proof. The tangential Hessian is \(\mathop{\mathrm{Hess}}f|_{TB}=(\partial_\nu f)\operatorname{II}_g\). Thus \(\mathop{\mathrm{tr}}_\perp H^f=w_\nu H\) and \(\chi q=F-P_B-w_\nu H\). The normal component of (44), using (51), is \[V_\nu/u=4d\partial_\nu t+\chi w_\nu q =4d\partial_\nu t+w_\nu(F-P_B)-vH.\] The \(\bar g\)-normal is \(\sqrt d\,\nu\). Its mean curvature is \(\sqrt d\,H\), because the tangential metric is unchanged and its normal first variation is multiplied by \(\sqrt d\). The conformal mean-curvature formula gives the asserted mean curvature for \(\hat g\). ◻

Proof. Fix a smooth \(j:\mathbb R\to[0,1]\) which vanishes for \(t\le0\) and equals one for \(t\ge1\). For \(0\le\lambda\le1\) put \[V_\lambda=uA_\chi(4\nabla Z+\lambda K(w,\cdot)),\qquad \mathcal B=-H+w_\nu(F-P_B)-N_Bd.\] Before the last stage use the boundary condition \[ \begin{split} (V_\lambda)_\nu/u ={}&[1-(1-\lambda)j(t)]\\ &\cdot[\mathcal B-(1-\lambda)(A_\chi K(w,\cdot))_\nu]. \end{split} \tag{56}\] Outer values remain \(f=Z=0\) throughout.

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

Choose the magnitudes larger than \(2\sup_B|H|+1\). Equation (37) then gives, for \(F=h+C+D_0\) at the assigned boundary height \(b=b_\pm\), \[ F(x,+\nu,b)\ge P_B+H,\qquad F(x,-\nu,b)\le P_B-H. \tag{57}\] These are limiting directional evaluations at \(a=1,\chi=0\). The compatible direction has equality; \(D_0\) corrects the other one. Its derivatives on bounded height ranges are independent of \(N\). Choose a smooth compactly supported extension \(b(x)\), constant equal to \(b_\pm\) on each smaller collar. On these collars let \(D_1(x,w)=A_1w\cdot\nu_s\), extended by a collar cutoff, with \(A_1>\sup_B|H|+1\). Then \(D_1(x,0)=0\) and \(D_1(x,\pm\nu)=\pm A_1\) on \(B\).

The four stages are the following.

1. For \(0\le s\le1\), set \(\lambda=1-s\), keep \(F=h+C\) and \(h|_B=b\), and use \(\mathop{\mathrm{div}}V_\lambda=u\Xi\) and (56).

2. For \(1\le s\le2\), set \(\lambda=0\), \(\alpha=s-1\), retain \(h|_B=b\), and replace the trace right side by \(F=h+C+\alpha D_0(x,w,h)\). Keep the same scalar equation and boundary prescription.

3. For \(2\le s\le3\), set \(\lambda=0\), \(\zeta=s-2\), prescribe \(h|_B=(1-\zeta)b\), and use \[ \begin{split} F={}&h+(1-\zeta)[C+D_0(x,w,h+\zeta b(x))]\\ &+\zeta[\mathop{\mathrm{tr}}_{A_\chi}K+D_1(x,w)]. \end{split} \tag{58}\] Keep the same scalar equation and boundary prescription. At a face, \(h+\zeta b=b\). The value at \(w=\pm\nu\) is the convex combination of the preceding directional value and \(P_B+D_1(x,\pm\nu)\), so (57) persists.

4. For \(3\le s\le4\), retain the trace problem at \(\zeta=1\) and multiply both \(\Xi\) and the right side of (56) by \(\eta_s=4-s\).

The prescriptions agree at the common endpoints. Their \(h\)-derivatives are \(1\), \(1+\alpha\partial_hD_0\), or \(1+(1-\zeta)\partial_hD_0\), all at least one.

At \(\zeta=1\) the trace equation is \(\mathop{\mathrm{tr}}_{A_\chi}H^f=h+D_1(x,w)\) with zero Dirichlet values. At a positive interior maximum of \(f\), one has \(w=0\), \(A_\chi=I\), \(D_1=0\), and \(l\Delta f=\tau f>0\), a contradiction. A negative minimum is excluded in the same way. Thus \(f=0\) and \(t=Z\) in the last stage. At \(s=4\) the remaining equation is \(\mathop{\mathrm{div}}(4L(Z)\nabla Z)=0\), with zero inner conormal flux and \(Z=0\) on \(S_R\). Multiplication by \(Z\) and integration give \(Z=0\). For the fixed-point construction in Section 6, the final linear scalar problem has zero source and zero inner flux for every input; consequently that last map has output identically zero. ◻

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

Proof. Take \(r_0\) outside the supports of \(K,C,D_0,D_1\). There \(F=h\) in the first three stages, and \(h=0\) in the last one. For \(\psi=r^{-\gamma}\), with axis \(\nabla\psi/|\nabla\psi|\), the controlled end gives, uniformly for \(0<\chi\le1\), \[\mathop{\mathrm{tr}}_{A_\chi}\mathop{\mathrm{Hess}}\psi =\gamma((\gamma+1)\chi-2)r^{-\gamma-2} +O(r^{-\gamma-3})<0\] after enlarging \(r_0\). At a comparison contact, coefficients use the given \(Z\) and the common gradient of test and solution. The positive factor \(l/(\tau\sqrt D)\) preserves this sign. Height monotonicity therefore makes \(C(\psi-R^{-\gamma})\) an upper barrier and its negative a lower barrier. Choose \(C\) to dominate \(C_0\) on \(r=r_0\) for every \(R\ge2r_0\). Differentiating at the common outer value gives (60); tangential derivatives there vanish.

At \(S_R\), \(Z=0\). The monotonicity of its defining expression implies \(t>-\epsilon\) if \(\ell^2|\nabla f|^2<e^{8\epsilon}-1\). It suffices to require \(C^2\ell^2\tau^{-2}R^{-2-2\gamma}<e^{8\epsilon}-1\), which is a permissible lower bound depending on fixed \(N,\epsilon\). ◻

Proof. Consider the graph of \(f\) in the Riemannian product with warped fiber \((\Omega\times\mathbb R,g+l^2\,\mathrm db_0^2)\). Its induced metric is \(\bar g\). At \(\,\mathrm dt=0\) one also has \(\,\mathrm dl=0\), and its second fundamental form, up to orientation, is \(H^f\). Write \(E=l^{-1}\partial_{b_0}\). The graph unit normal is \(\sqrt d\,E-ae\). The ambient curvature formulas at this point are \[\begin{gathered} \mathop{\mathrm{Scal}}_{\mathrm{amb}}=\mathop{\mathrm{Scal}}_g-2\Delta l/l,\\ \mathop{\mathrm{Ric}}_{\mathrm{amb}}|_{T\Omega}=\mathop{\mathrm{Ric}}_g-\mathop{\mathrm{Hess}}l/l,\qquad \mathop{\mathrm{Ric}}_{\mathrm{amb}}(E,E)=-\Delta l/l, \end{gathered}\] with zero mixed Ricci terms. Hence the ambient contribution to half the graph scalar curvature is \[\tfrac12\mathop{\mathrm{Scal}}_g-v\mathop{\mathrm{Ric}}_g(e,e) -v\,\mathop{\mathrm{tr}}_\perp\mathop{\mathrm{Hess}}l/l.\] The last term is nonpositive: at \(\,\mathrm dt=0\), \(\mathop{\mathrm{Hess}}l=lp\,\mathop{\mathrm{Hess}}t\) and \(p\ge0\). The second-form contribution is \([(\mathop{\mathrm{tr}}_{\bar g}H^f)^2-|H^f|_{\bar g}^2]/2\). Since the inverse graph metric is \(A_d\), its expansion is exactly (62). Finally the conformal contribution at this point is \(-4\bar\Delta t\le0\). This proves (61).

The pointwise algebraic clause of (OpenAI 2026b, Lemma 4.3) applies under Equation (47). We also record its three-dimensional calculation, because the boundary homotopy below needs its precise degenerate normal coefficient. First evaluate \(\mathcal S\) on \(S=K+H^f\), whose trace in the transverse plane is \(F-\chi q\). Subtracting (62) from (46) at \(\,\mathrm dt=0\) gives \[ \begin{split} \mathcal T-\mathcal S(S) ={}&|T^{\operatorname{tf}}|^2+(1+d)|M|^2\\ &+\chi(1+d-\chi/2)q^2-dFq+F^2/2. \end{split} \tag{65}\] The normal coefficient is at least \(\chi\), while \(|dFq|\le\chi q^2/2+d^2F^2/(2\chi)\). Because \[d^2/\chi=d(4+pv)/4\le1+N/4,\] Equation (49) implies \(\mathcal T-\mathcal S(S)\ge c\mathcal T-C(1+N)F^2\) with universal \(c>0\).

To replace \(S\) by \(H^f=S-K\), polarize (62). The difference is bounded by \[C|K|(|T^{\operatorname{tf}}|+|M|+|F|+d|q|)+C|K|^2.\] Indeed the only extra normal product is \(dq\,\mathop{\mathrm{tr}}_\perp K\); all remaining products involve transverse entries, \(dM\), or \(\mathop{\mathrm{tr}}_\perp S=F-\chi q\). By (49) and \(d/\sqrt\chi\le C\sqrt{1+N}\), this bound is at most \(C\sqrt{1+N}|K|\sqrt{\mathcal T}+C|K|^2\). Young’s inequality absorbs a fixed small fraction of \(\mathcal T\) and costs only \(C(1+N)|K|^2\). This proves (63).

If \((1+p)v\le\eta_*\), then \(d\) and \(\chi\) are within \(C\eta_*\) of one. At \(d=\chi=1\) the normal block in (65) is \(3q^2/2-Fq+F^2/2\); its symmetric matrix has smallest eigenvalue \((2-\sqrt2)/2>0\). Choose \(\eta_*>0\) small enough that its coefficients remain within half this spectral gap. The transverse coefficients are already bounded below. For \(K=0\), Equation (49) therefore gives \(\mathcal T-\mathcal S(H^f)\ge c_*\mathcal T\) in this regime. In the remaining regime \(v>\eta_* /(1+p)\), use (63) with \(K=0\) and absorb \((1+N)F^2\) into \(C(1+N)^2vF^2\). This proves (64). All formulas extend to \(\sigma=0\) by their invariant expressions; there \(d=\chi=1\) and the full quadratic expressions are independent of the temporary choice of axis. ◻

Proof. At this contact \(\,\mathrm dt=0\) and \(p=N\) is locally constant as a function of \(t\) near its value \(-\epsilon\). Differentiating \(w\) in an orthonormal frame gives \[ \nabla_iw_j=H^f_{ij}-w_jH^f_{ik}w^k. \tag{67}\] In the \(e\)-frame the normal vector slot is multiplied by \(d\). Equations (49) and \(d/\sqrt\chi\le C\sqrt{1+N}\) therefore show \[ |\nabla w|\le C(1+N)^A(|K|+\sqrt{\mathcal T}). \tag{68}\] No bound for the uncontracted normal Hessian is asserted.

Each first-three-stage right side can be written as a smooth function \(F(x,w,h,p)\). Its \(h\)-derivative is at least one. On the bounded height range supplied by Lemma 16, its explicit \(x\)- and \(w\)-derivatives are polynomially bounded and compactly supported, apart from the derivative of the term \(h\). This follows directly from the fixed collar cutoffs and \(A_\chi=I-(p+4)w\otimes w/(4+pv)\), whose denominator is at least four. For example its first \(w\)-derivatives on \(|w|\le1\) are bounded by \(C(1+N)^2\), and the other explicit modifications have bounds independent of \(N\). Combining these bounds with (68) and Young’s inequality therefore produces a fixed finite power of \(1+N\). Since \(\nabla p=0\) at the contact and \(w(h)=\tau a\sigma\), the chain rule and (68) give, for every fixed \(\eta>0\), \[ w(F)\ge\tau a\sigma-\eta\mathcal T -C_\eta(1+N)^A\mathbf1_{\mathcal K}. \tag{69}\] The dependence \(D_0(x,w,h+\zeta b(x))\) only adds the fixed compact derivative of \(b\) and does not change this conclusion.

The omitted flux is \((1-\lambda)uA_\chi K(w,\cdot)\). Let \(B_K=A_\chi K(w,\cdot)\). Its derivative is bounded by a polynomial constant times \(1+\sqrt{\mathcal T}\) on \(\mathcal K\), by (68); derivatives of \(p\) vanish. The weight term requires the contraction rather than a bound for \(|\nabla\log u|\). At the contact, \[\nabla\log u=H^f(w,\cdot),\qquad \langle\nabla\log u,B_K\rangle =a^2\bigl(\chi H^f_{ee}K_{ee} +H^f_{e\perp}\cdot K_{e\perp}\bigr).\] This involves only \(\chi H^f_{ee}\) and \(H^f_{e\perp}\) and is controlled by (49). It follows that \[ |\mathop{\mathrm{div}}(uB_K)|/u \le\eta\mathcal T+C_\eta(1+N)^A\mathbf1_{\mathcal K}. \tag{70}\]

Combine the scalar identity with (61). The difference \(8\pi\mu-\mathop{\mathrm{Scal}}_g/2\) equals \(((\mathop{\mathrm{tr}}K)^2-|K|^2)/2\) and is compactly supported; so is \(J\). The term \(F\mathop{\mathrm{tr}}K\) is compactly bounded by Lemma 16. On \(\mathcal K\), Equation (63) therefore contributes \(c_*\mathcal T-C(1+N)^A\). Outside \(\mathcal K\), \(K=0\), \(F=h\), and (64) together with Lemma 17 gives \(c_*\mathcal T-C(1+N)^Av r^{-2\gamma}\). The remaining Ricci term is bounded below by \(-C\mathbf1_{\mathcal K}-Cv r^{-3}\); enlarge \(\mathcal K\) if necessary. Both end weights are bounded by a multiple of \(\rho_0+\rho_1\). Finally subtract the omitted flux and use (69)–(70), choosing their two \(\eta\)’s with sum at most \(c_*/2\). Fix \(0<\delta_0<\min(c_*/4,1/4)\). Decreasing it slightly if necessary gives (66). ◻

Proof. Choose \(R_{\mathrm{floor}}\) to include the requirement of Lemma 17. A floor contact cannot occur on \(S_R\). In the first three stages, an inner contact has \(j(t)=0\). Equation (56) then cancels the omitted flux on both sides and gives \(V_\nu/u=\mathcal B\). Equation (52) forces \(H+4\partial_\nu t=-N_B\), so \(\partial_\nu t<0\). This contradicts the nonnegative derivative into \(\Omega_R\) at a boundary minimum.

Consider an interior contact. Let \(E_N=C(1+N)^A[\mathbf1_{\mathcal K}+v(\rho_0+\rho_1)]\) denote the error in (66). Because \(\rho_0\) has a positive minimum on \(\mathcal K\), \(\mathbf1_{\mathcal K}\le C_{\mathcal K}\rho_0\). Also \(v\rho_0\le\rho_0\), and \(\rho/\rho_0\) is globally bounded. We can therefore choose a polynomial \(C_N\) so large that, pointwise for all \(0\le v\le1\), \[ \mathcal P\ge E_N+2\rho+\rho_0. \tag{71}\] At the floor \(m_0=1\), so the scalar equation and the lower estimate give \[\begin{split} 0 &\ge \delta_0\mathcal T+(1-\delta_0)\tau a\sigma +\mathcal P-E_N-\rho\\ &\ge \rho+\rho_0>0, \end{split}\] a contradiction. Enlarging \(C_N\) below preserves this inequality and preserves its polynomial dependence.

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

Proof. For integers \(j\ge1\) define \[ S_j(s)=\bigl(1+\exp[-j^2(s-k-j^{-1})]\bigr)^{-1}, \qquad v_j=S_j\circ u. \tag{73}\] These functions are lower semicontinuous and take values in \((0,1)\). Their lower relaxed limit \(\liminf_{j\to\infty,\,y\to x}v_j(y)\) is the function \(I_k(x)\) equal to zero on \(D_k\) and one on its complement. Indeed, at \(x\in D_k\) the constant sequence gives \(v_j(x)\le S_j(k)\to0\), while \(v_j\ge0\) everywhere. If \(u(x)>k\), lower semicontinuity gives a neighborhood on which \(u\ge k+\delta\) for some \(\delta>0\), and \(v_j\to1\) uniformly there.

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

For each fixed large \(j\), restrict to a neighborhood of \(x_j\) on which \(\phi_0+a_j\) takes values in \((0,1)\). The increasing function \(S_j^{-1}\) makes \(S_j^{-1}(\phi_0+a_j)\) a smooth lower test of \(u\). Its contact height is \[u(x_j)=k+j^{-1} +j^{-2}\log\frac{v_j(x_j)}{1-v_j(x_j)} \le k+j^{-1}<k'\] for all sufficiently large \(j\). Its gradient is nonzero, since \(\,\mathrm d\phi_0(x_j)\to\,\mathrm d\phi(x)\ne0\). An increasing smooth composition preserves the oriented level expansion: it preserves the unit normal, and for \(\theta'>0\), \[\mathop{\mathrm{Hess}}(\theta\circ\phi_0) =\theta'\mathop{\mathrm{Hess}}\phi_0 +\theta''\,\mathrm d\phi_0\otimes\,\mathrm d\phi_0, \qquad \frac{\mathop{\mathrm{tr}}_{(\,\mathrm d\phi_0)^\perp}\mathop{\mathrm{Hess}}(\theta\circ\phi_0)} {|\,\mathrm d(\theta\circ\phi_0)|} =\frac{\mathop{\mathrm{tr}}_{(\,\mathrm d\phi_0)^\perp}\mathop{\mathrm{Hess}}\phi_0}{|\,\mathrm d\phi_0|}.\] The hypothesis therefore gives \(\mathcal E_Q[\phi_0+a_j](x_j)\le q_0\). Letting \(j\to\infty\) yields \(\mathcal E_Q[\phi](x)\le q_0\), because \(\phi_0\) and \(\phi\) have the same two-jet at \(x\). ◻

Proof. We prove the lower assertion. It is enough to consider an arbitrary sequence \(N_i\to\infty\), \(R_i\to\infty\) and solutions in either of the first two stages. The elementary height and end estimates in Lemmas 16 and 17 give \[ |h_i|\le C_0,\qquad |h_i|\le C(r^{-\gamma}-R_i^{-\gamma})\quad\hbox{on the end}, \qquad 3/4<\gamma<1. \tag{75}\] Near each white face extend \(h_i\) to the other side by the constant \(b_+\). Extend the fixed geometric coefficients smoothly there. The extended functions are continuous. Define the lower relaxed limit \[\underline h(x)=\lim_{j\to\infty} \inf\{h_i(y):i\ge j,\ \operatorname{dist}(x,y)<1/j\}.\] It is lower semicontinuous, bounded, and satisfies the limiting end bound. No equicontinuity of \(h_i\) is asserted.

Choose \(b_-<k'<\min\{0,b_-+\eta\}\), where \(\eta\) is the low-height sign range of the added term \(D_0\). Suppose that a smooth function \(\phi\) touches \(\underline h\) from below at \(x\), with \(\underline h(x)\le k'\) and \(d\phi(x)\ne0\). Exclude for the moment the black faces. Subtracting a small fourth-order localization term from \(\phi\) makes the contact strict without changing its two-jet. Minimization on a small closed ball then produces, after passing to a subsequence, points \(x_i\to x\) and constants \(c_i\to0\) such that \(\phi+c_i\) touches \(h_i\) from below at \(x_i\), and \(h_i(x_i)\to\underline h(x)\). The boundary of this ball cannot contain \(x_i\) by strictness. Nor can \(x_i\) lie on a white face or on the extended side: their values there are \(b_+>k'\). Thus every sufficiently large \(i\) uses the interior trace equation.

At these contacts, \[\nabla f_i=\tau_i^{-1}\nabla\phi, \qquad d_i\le\frac{\tau_i^2}{\ell_i^2|d\phi(x_i)|^2} \longrightarrow0, \qquad a_i\longrightarrow1, \qquad \chi_i\longrightarrow0.\] Here \(a_i=l_i|\nabla f_i|/\sqrt{D_i}\) is the deformation variable. Positivity of \(A_{\chi_i}\) and the Hessian inequality at a lower contact imply \[\mathop{\mathrm{tr}}_{A_{\chi_i}}K+ \frac{l_i}{\tau_i\sqrt{D_i}} \mathop{\mathrm{tr}}_{A_{\chi_i}}\mathop{\mathrm{Hess}}\phi \le F_i(x_i).\] In the second stage \(D_0\le0\) at these values; in the first stage it is absent. Consequently \(\limsup F_i(x_i)\le k'+C_b\). Passing to the limit gives \[ \frac{\mathop{\mathrm{tr}}_{(d\phi)^\perp}\mathop{\mathrm{Hess}}\phi}{|d\phi|} +\mathop{\mathrm{tr}}_{(d\phi)^\perp}K\le k'+C_b. \tag{76}\] This is the expansion of the level surface of \(\phi\), oriented toward increasing \(\phi\). The only Hessian used in this passage is the fixed test Hessian. Removing the localization proves the assertion for non-strict contacts as well.

Fix \(b_-<k<k'\) and put \(E_k=\{\underline h\le k\}\). Apply Lemma 21 with \(u=\underline h\), tensor \(Q=K\), and \(q_0=k'+C_b\). The function is finite and lower semicontinuous, and Equation (76) supplies the required bound for every nonzero-gradient lower test below \(k'\). We apply the lemma locally away from black faces, using the constant extensions to obtain open neighborhoods across white faces. Every exterior support of \(E_k\) there consequently has expansion at most \(k'+C_b\), with no regularity assumption on the sublevel.

This argument also rules out a hidden layer at a white face. At every point strictly on the added side, the relaxed limit equals \(b_+>k\). Thus \(E_k\) lies locally in the original closed exterior, and the reversed white face is an exterior support of \(E_k\) at any contact there, with expansion \[-H_B+P_B=-c_w>0.\] For \(k'\) sufficiently close to \(b_-\), this is strictly greater than \(k'+C_b\). The high extension ensured that all preceding low-value tests came from the interior equation, so the support contradiction applies at the face itself. Hence \(E_k\) avoids every white face. The end estimate makes \(E_k\) compact.

Adjoin the entire closed region \(\mathcal B_{c_b}\). At a point on its smooth black frontier, every exterior support of the union also contains the black region locally. The tangency comparison therefore bounds its expansion by \(c_b<k'+C_b\). At all other frontier points the preceding sublevel argument applies. The resulting compact closed set contains the required original inner barrier with a collar, because the black region already does. It avoids the outer barriers. Lemma 10 places it in a smooth black trapped region at any threshold \(c''>k'+C_b\) sufficiently close to \(c_b\). It is consequently contained in \(\mathcal B_{c''}\).

Finally let \(Q\) be as in the lemma. Hausdorff right continuity at \(c_b\) gives a \(\delta>0\) such that \(\mathcal B_{c_b+\delta}\cap Q=\varnothing\); decrease \(\delta\) so that the enlarged threshold is still negative and all barriers remain strict. Choose \(k,k',c''\) with \[b_-<k<k',\qquad k'+C_b<c''<c_b+\delta.\] The construction implies \(E_k\cap Q=\varnothing\) for every sequence under consideration. If the uniform estimate \(h>k\) on \(Q\) failed, choose violating solutions with \(N_i\to\infty\), \(R_i\to\infty\) and \(x_i\in Q\), \(h_i(x_i)\le k\). A subsequence has \(x_i\to x\in Q\) and \(\underline h(x)\le k\), a contradiction. This gives (74), after decreasing its positive margin.

For the upper assertion apply the entire argument to \((-h,-K,-C)\), interchanging colors. The reversed black faces have strictly positive white expansion \(-c_b\). Their exclusion and compactness give positive distance from the black region before applying the white version of Lemma 10. Adjoining the full white region is therefore legitimate in the fixed black complement. Empty white regions and missing face types cause no change: only an actually nonempty sublevel would supply a trapped seed and hence a contradiction. ◻

Proof. Write \(s\) for the smooth leaf parameter in an outward collar and \(\nu_s=\nabla s/|ds|\). All geometric coefficients of this fixed collar are bounded, with \(|ds|\) bounded above and below. At a comparison contact with \(h_0=b_-+\psi(s)\), \(\psi'>0\), the shared gradient gives \[ d\le C\frac{\tau^2}{\ell^2(\psi')^2},\qquad 1-a\le d,\qquad N\chi\ge c_1d\quad(N\ge4). \tag{78}\] If the slopes are in a fixed positive interval, the first bound is \(d\le C\ell\). None of these comparisons uses an upper bound for \(Z\).

The trace operator on this test is \[\begin{align*} &\mathop{\mathrm{tr}}_{A_\chi}\left(K+ \frac{l}{\tau\sqrt D}\mathop{\mathrm{Hess}}h_0\right)\\ &\quad=P_s+aH_s+\chi K(\nu_s,\nu_s) +a\chi\left( \frac{\mathop{\mathrm{Hess}}s(\nu_s,\nu_s)}{|ds|} +|ds|\frac{\psi''}{\psi'}\right), \tag{79}\end{align*}\] where \(P_s\) is the tangential trace of \(K\) on the leaf. Thus it is the leaf expansion \(H_s+P_s\), with an error of absolute value at most \(C d\), plus the displayed logarithmic-slope term. The same formula for a negative slope has \(a\) replaced by \(-a\) in the Hessian terms. This is an evaluation at the contact variables; it does not differentiate the unknown \(t\).

On a black collar \(H_s+P_s\ge c_b\) and \(C=C_b-k_Bs\). Choose small positive slopes so that \(\psi(0)=0\) and \(\psi(s)\le k_Bs/2\). At the outer end of a sufficiently short fixed collar, Lemma 22 lets this test lie below the solution. In the compatible direction the add-on \(D_0\) vanishes. The residual of the trace equation on the test is therefore at least \[ k_Bs/2-Cd+a\chi|ds|\frac{\psi''}{\psi'}. \tag{80}\] Choose a smooth nonnegative \(\beta_N(s)\) equal to \(AN\) for \(0\le s\le1/N\), supported in \(0\le s\le2/N\), and satisfying \(\int\beta_N\le2A\). Define \[\psi'(s)=m\exp\left(\int_0^s\beta_N(q)\,\mathrm dq\right).\] Choose the fixed \(A\) large enough that the final term of (80) dominates \(Cd\) on \(s\le1/N\), using (78) and \(a\ge1/2\) there. On \(s\ge1/N\), the first term dominates \(Cd\le C\ell\) for large \(N\); the taper contributes with the good sign. Since the slope multiplier is at most \(e^{2A}\), a fixed sufficiently small \(m>0\) enforces all the previous small-slope requirements. This constructs a strict lower barrier with slopes bounded above and below independently of \(N\).

An upper barrier from \(b_-\) is constructed in the same collar with \[\psi'(s)=M\exp\left(-\int_0^s\beta_N(q)\,\mathrm dq\right).\] The smooth leaf expansion differs from its boundary value by \(O(s)\). Choose the fixed \(M\) so large that \(\psi(s)\ge C_2s\) dominates this variation, the offset variation, and the height bound at the outer collar end. The logarithmic-slope term is now negative. On \(s\le1/N\) it dominates \(Cd\); on the rest of the collar the negative margin proportional to \(s\) does so. The upper barrier is strict at every possible positive contact. Its slopes lie in \([Me^{-2A},M]\), an interval independent of \(N\).

To check the comparison signs, at a negative interior minimum of \(h-h_0\) the Hessian of \(h\) dominates that of \(h_0\), the gradients and \(Z\) agree, and hence so do \(t,l,A_\chi\). The right side is strictly increasing in the height. Thus a strict positive residual for a lower barrier contradicts the equation. At a positive maximum the inequalities reverse, proving the upper-barrier assertion. Taking the one-sided derivative at \(s=0\) proves the black part of (77). Replacing \((h,K,C)\) by \((-h,-K,-C)\) proves the white part.

In the third stage the boundary height is the assigned interpolated constant, say \(b_\zeta\). Construct tests \(b_\zeta+\psi(s)\) and \(b_\zeta-\psi(s)\) with large positive \(\psi'\) and \((\log\psi')'=-\beta_N\). At \(s=0\), the two directional inequalities of Proposition 15 say precisely that the positive-slope test has nonpositive limiting residual and the negative-slope test nonnegative limiting residual. At every comparison contact, the fixed positive barrier slope and Equation (78) give \(d\le C\ell\), so \(a\ge1/2\) for large \(N\). On \(1/2\le|w|\le1\), the first \(w\)-derivatives of \(A_\chi\) in Equation (41) are bounded independently of \(0\le p\le N\). The other collar terms have fixed smooth coefficients on the bounded height range. Thus, at fixed boundary height the coefficient and geometric errors are \(O(s+d+\chi)\), since \(|w-(\pm\nu_s)|=1-a\le d\) and the radial comparison segment stays in this range. Changing the test height supplies, by height monotonicity, a residual of magnitude at least \(\psi(s)\) with the required sign. Large fixed slopes dominate the \(O(s)\) error and the outer-end height bound. The logarithmic-slope term has the required sign for both tests, as follows from (79); it dominates the \(O(d)\) error on the initial \(1/N\) interval. The remaining margin dominates it beyond that interval. This proves the two-sided upper derivative bound, without using sublevel exclusion in the third stage.

Finally, on each own-color collar the lower barrier just constructed gives the required one-sided height bound. On the rest of a fixed compact region use Lemma 22; the opposite height bound on an own-color collar follows from that lemma as well, since the closed collar is disjoint from the other region. A finite cover proves \(b_-\le h\le b_+\) on the compact regions in the statement. ◻

Proof. For \(\sigma>0\), write \(e_f=\nabla f/\sigma\), the axis denoted by \(e\) in Section 4. Set \[W=l\bar\nabla f=\frac{l\nabla f}{D}=\sqrt d\,a e_f.\] Its norm in \(\bar g\) is \(a\le1\). For any covector \(\xi\), \[ |\xi(W)|\le |\xi|_{A_d} \le\sqrt{1+N/4}\,|\xi|_{A_\chi}. \tag{83}\] Direct differentiation, using \(uW=Lw\), gives \[\begin{align*} \frac{\mathop{\mathrm{div}}(uW)}u &=\sqrt d\left[ \mathop{\mathrm{tr}}_{A_d}H^f+(dp+p_L)a\partial_{e_f}t\right], \tag{84}\\ \frac{\mathop{\mathrm{div}}(\sqrt D\,W)}{\sqrt D} &=\sqrt d\left[ \mathop{\mathrm{tr}}_{A_d}H^f+dp\,a\partial_{e_f}t\right]. \tag{85}\end{align*}\] For example, \(\mathop{\mathrm{div}}w=\mathop{\mathrm{tr}}_{A_d}H^f+dp\,a\partial_{e_f}t\), while differentiating \(L\) adds \(p_La\partial_{e_f}t\) to the first formula. Coercivity in Lemma 13 bounds both right sides by \(\Pi_N(1+\sqrt{\mathcal T})\). More explicitly, writing \(q=(K+H^f)_{e_fe_f}\), the normal Hessian term has coefficient \(d^{3/2}\), so it is bounded by \(\sqrt d|q|+C\). The other Hessian terms are transverse traces, and the gradient term is at most \((2N+2)\sqrt d|\partial_{e_f}t|\). All are controlled by the stated coercivity estimate. This also establishes the formulas and bounds at zero gradient by their smooth direction-free extensions.

Let \(\varepsilon_B=1\) on a black collar and \(-1\) on a white collar. The signed slopes give \(w_\nu=\varepsilon_B a\) on \(B\) and \(a\ge1/2\) for large \(N\). Thus \[uW_\nu=\varepsilon_B La, \qquad \sqrt D\,W_\nu=\varepsilon_B a.\] The outward integration normal is \(-\nu\). Apply the divergence theorem to \(-\varepsilon_B uW\varphi\), with a fixed collar cutoff. Its boundary flux is \(La\varphi\). Equations (83) and (84), and the fixed cutoff derivative, give (81). Replace \(u\) by \(\sqrt D\) and use (85) for (82). No factor \(\ell^{-1}\) was used. ◻

Proof. Extend the fixed base patch smoothly and embed a compact enlargement isometrically in Euclidean space by Nash’s embedding theorem (Nash 1956, Theorem 2, p. 59). Its product with the vertical line gives an isometric Euclidean immersion of the graph. The second fundamental form of the base embedding is bounded, so its contribution to the graph mean curvature vector is bounded. The scalar mean curvature in the product is \[H_\Gamma= \frac{\sqrt D}{l\sqrt{1+\sigma^2}} \left(\mathop{\mathrm{tr}}_{e_f^\perp}H^f+d_0H^f(e_f,e_f)\right).\] The prefactor is bounded by \(C(N)\), and \(d_0/\sqrt d=\sqrt{1+l^2\sigma^2}/(1+\sigma^2)\le e\). Coercivity therefore gives \(|\boldsymbol H_\Gamma|\le C(N)(1+\sqrt{\mathcal T})\). This estimate involves controlled quadratic energy, not a pointwise bound for the Hessian.

The ordinary Michael–Simon inequality (Michael and Simon 1973), in the compact-support form stated in (Simon 2014, chap. 4, Section 5, Theorem 5.7, p. 98), gives after the boundary extension described below, for nonnegative \(\varphi\), \[\left(\int_\Gamma\varphi^{3/2}\,\mathrm d\Gamma\right)^{2/3} \le C\left[ \int_\Gamma(|\nabla_\Gamma\varphi| +|\boldsymbol H_\Gamma|\varphi)\,\mathrm d\Gamma +\int_{\partial\Gamma}\varphi\,\mathrm dA\right].\] One can obtain the boundary form directly from the compact-support version by smoothly extending each smooth graph across its boundary and cutting off on a collar tending to zero: the cutoff derivative integral converges to the boundary integral, and the added curvature integral tends to zero. This argument is applied to each smooth graph before the uniform estimate is taken. The only boundary in the support is an inner face, where \(f\) is constant and \(\,\mathrm dA=\,\mathrm dA_g\). Equation (82) and the fixed-\(N\) graph comparisons bound its contribution by \[C(N)\int_\Gamma [(1+\sqrt{\mathcal T})\varphi +|\nabla_\Gamma\varphi|]\,\mathrm d\Gamma.\] Apply the resulting inequality to \(\varphi=|\omega|^4\). Cauchy–Schwarz bounds its right side by \[C(N)\|\omega\|_6^3 \left(\int_\Gamma [|\nabla_\Gamma\omega|^2+(1+\mathcal T)\omega^2] \,\mathrm d\Gamma\right)^{1/2}.\] The left side is \(\|\omega\|_6^4\); division, with the zero case separate, and squaring prove the assertion. ◻

Proof. It remains to extract the bounds for the conformal variable and the gradient. Since \(Z=t+\log D/8\) and \(D\ge1\), the floor gives \(-\epsilon\le t\le Z\). The upper bound for \(Z\) gives \[\sigma^2=\frac{e^{8(Z-t)}-1}{l^2} \le\ell^{-2}\bigl(e^{8(C(N)+\epsilon)}-1\bigr).\] This proves the asserted fixed-\(N\) gradient bound. The height bounds are Lemma 16. The finite collection of sublevel and collar estimates and the polynomial absorption choose \(N_0\) once. Enlarge \(R_{\min}(N)\) to include the outer estimates and all fixed compact patches used above. The next section derives Hessian and continuity estimates from these bounds and uses them to prove existence. ◻

Proof. We first obtain an \(L^p\) Hessian estimate for some \(p>2\). A divergence identity then shows that a gradient nearly attaining its upper bound forces the solution to be nearly affine. The equation with a fixed gradient direction improves that affine approximation. The same steps survive reflection at a constant Dirichlet face, and iteration gives the stated estimate.

Proof. The proof has two steps. An exponential change absorbs the quadratic term in \(DZ\); a small potential correction then permits weak Harnack to decrease the essential oscillation. Both steps work in the two dimensions stated in the lemma. Restricting all radii to at most one, we may replace \(\beta\) by \(\min\{\beta,1\}\) in the measure bound. We retain the symbol \(\beta\) for this positive reduced exponent; this also makes the small-scale mollification bound below immediate.

Write \(\mathcal L=-\operatorname{div}(aD)\) and, for \(s\in\{-1,1\}\), set \(V_s=e^{skZ}\). The assumed product rule is \[ \mathcal L V_s =\operatorname{div}(skV_sb)-skV_se -k^2V_s\bigl(aDZ\cdot DZ+b\cdot DZ\bigr). \tag{129}\] The term \(V_se\) uses the fixed bounded representative. For a continuous piecewise smooth function the identity follows by integration on the two sides; its common trace multiplies the flux jump. Choose \(k\ge\max\{1,4C_0/\lambda\}\) and use \(B|DZ|\le(\lambda/2)|DZ|^2+B^2/(2\lambda)\). The gradient-square term from \(e\) is absorbed by the negative quadratic term for both signs. Since \(e^{-kM}\le V_s\le e^{kM}\), this gives \[ \mathcal L V_s\le\operatorname{div}F_s+\nu, \qquad |F_s|\le C, \qquad \nu=C(\,\mathrm dx+\mu)\ge0. \tag{130}\] Thus no small-ball estimate for \(|DZ|^2\) is needed.

On a ball \(B_R\) in the smaller patch solve the zero-Dirichlet problem \(\mathcal L w_s=\operatorname{div}F_s+\nu\). For the bounded vector source, the scalar bounded-solution estimate with any fixed \(p>n\) gives \[\|w_{F_s}\|_\infty \le CR^{1-n/p}\|F_s\|_{L^p(B_R)}\le CR;\] this is the scaled divergence-source estimate of (Littman et al. 1963, Theorem 2.6, p. 50). For the positive measure source, extend \(a\) elliptically to \(B_{2R}\). The scalar Green comparison theorem (Littman et al. 1963, Theorem 7.1 and Remark 2, p. 66), and domain monotonicity, give \[0\le\mathcal G_{B_R}(x,y)\le\mathcal G_{B_{2R}}(x,y) \le C|x-y|^{2-n},\qquad x,y\in B_R.\] Using \(B_{2R}\) keeps both points in a fixed compact interior region of the comparison ball after rescaling. Dyadic annuli and (128) imply \[ \begin{split} \sup_{x\in B_R}\int_{B_R}\mathcal G_{B_R}(x,y)\,\mathrm d\nu(y) &\le C\sum_{j\ge0}(2^{-j}R)^{2-n} \nu(B_{2^{1-j}R}(x))\\ &\le C(R^2+R^\beta). \end{split} \tag{131}\] All balls used in this estimate lie in the chosen larger patch.

This potential also gives an energy solution. Mollify the restriction of \(\nu\) to \(B_R\) and restrict the resulting positive density again to \(B_R\). Its ball-growth bounds are uniform: above the smoothing scale use an enlarged ball, and below it use the mollified density bound. The variational solutions have the preceding supremum bound and obey \[\lambda\int|Dw_j|^2\le\int w_j\,\mathrm d\nu_j \le\|w_j\|_\infty\nu_j(B_R).\] Weak compactness in \(W^{1,2}_0\) and distributional convergence produce the desired measure-source solution. Adding the vector-source solution, and taking \(R\le1\), yields \[ \|w_s\|_\infty\le\varepsilon_R, \qquad\varepsilon_R=C(R+R^\beta). \tag{132}\]

All extrema in the rest of this proof are essential extrema. Put \(M_R=\operatorname*{ess\,sup}_{B_R}Z\) and \(m_R=\operatorname*{ess\,inf}_{B_R}Z\). The functions \[ H_+=e^{kM_R}-e^{kZ}+w_++\varepsilon_R, \qquad H_-=e^{-km_R}-e^{-kZ}+w_-+\varepsilon_R \tag{133}\] are nonnegative almost everywhere and satisfy \(\mathcal L H_\pm\ge0\). At least half of \(B_{R/2}\) satisfies either \(Z\le(M_R+m_R)/2\) or \(Z\ge(M_R+m_R)/2\). On the first set the positive exponential deficit is at least \(c(M_R-m_R)\); on the second set the negative exponential deficit has that lower bound. Their derivatives are bounded above and away from zero on \([-M,M]\). Weak Harnack for the corresponding \(H_\pm\) therefore either lowers the essential supremum or raises the essential infimum on \(B_{R/4}\) by at least \(c'(M_R-m_R)-C\varepsilon_R\). Hence \[ \operatorname*{ess\,osc}_{B_{R/4}}Z \le(1-c')\operatorname*{ess\,osc}_{B_R}Z+C(R+R^\beta). \tag{134}\] Choose a positive exponent smaller than \(\min\{1,\beta\}\) and than \(-\log(1-c')/\log4\). Iteration gives an essential oscillation bound \(Cr^\gamma\) at every center in a smaller patch. The averages of \(Z\) on balls centered at \(x\) are consequently Cauchy as their radii decrease to zero. Their limit defines \(Z^*(x)\) at every center; comparison on overlapping balls gives \(|Z^*(x)-Z^*(y)|\le C|x-y|^\gamma\). Lebesgue differentiation identifies \(Z^*=Z\) almost everywhere. This proves the representative assertion. For continuous \(Z\) the representatives agree everywhere. No conclusion about arbitrary values of the initially chosen Borel representative on a null set is required.

For the boundary application, flatten a face and set \(S=\operatorname{diag}(1,\ldots,1,-1)\). An even extension uses \(a_-=Sa_+S\) and \(b_-=Sb_+\); its plane source is twice the prescribed total normal flux. An odd extension after subtracting a constant Dirichlet value uses \(a_-=Sa_+S\), \(b_-=-Sb_+\) and the sign-reflected bulk source; its normal flux is continuous. The coordinate volume density is included in these coefficients. For the smooth reflected solutions used here the product rule holds, and a bounded plane density has ball mass \(O(r^{n-1})\), giving \(\beta=1\) for that part of the measure. This verifies the boundary clause whenever the remaining reflected measure has the stated growth. ◻

Proof. First, control \(Df\) and the Hessian measure. Proposition 26 bounds \(f,Z,t\) and \(\sigma=|\nabla f|\). At fixed parameters this bounds \(u,l,D\) above and away from zero and gives \(\chi\ge\chi_0>0\). After dividing the scalar trace equation by \(l/\sqrt D\), its right side is \[ G_f=\frac{\sqrt D}{l} \bigl(F_s-\mathop{\mathrm{tr}}_{A_\chi}K\bigr). \tag{135}\] It is bounded throughout the homotopy, including its added collar terms. Lemma 27 therefore gives a uniform \(C^{1,\alpha}\) estimate for \(f\) and \[ \int_{B_r}|\mathop{\mathrm{Hess}}f|^2\le Cr^{1+2\alpha} \tag{136}\] on the uniform patches, including boundary patches where \(f\) is constant.

Second, control the oscillation of \(Z\). After multiplication by the Riemannian volume density, the second equation has coordinate principal coefficients \(a^{ij}=4\sqrt{\det g}\,uA_\chi^{ij}\), a bounded flux vector \(b\), and a source bounded in absolute value by \[ C\bigl(1+|DZ|^2+|D^2f|^2\bigr). \tag{137}\] Indeed \(t\) is a smooth function of \(x,Z,Df\) on the bounded variable range. Thus \(Dt\) is a bounded linear combination of \(DZ,D^2f\) and fixed smooth terms; the expression \(\mathcal T\) is quadratic in these quantities. All other source terms have lower order. The same volume density is included in \(b,e\); it is smooth and bounded above and away from zero on the chosen patches, so it preserves these bounds and uniform ellipticity. The prescribed conormal flux at the inner boundary is bounded. At the outer face \(Z=0\). Apply Lemma 28, with \(\mu=C|D^2f|^2\,\mathrm dx\) and the bounded reflected plane density. Equation (136) supplies its measure growth hypothesis. We obtain a uniform positive Hölder exponent for \(Z\).

Third, gain two derivatives of \(f\). The coefficients and source of the scalar equation are smooth functions of \(x,Z,Df\) on the bounded range. They are therefore Hölder now that \(Z\) and \(Df\) are Hölder. Interior and constant-Dirichlet boundary Schauder estimates give \(f\in C^{2,\theta}\) for a uniform \(\theta>0\). Expand the \(Z\)-equation in nondivergence form. Its principal coefficients are uniformly elliptic and \(C^{0,\theta}\), while its right side is bounded by \(C(1+|DZ|^2)\). On an inner face \(Df\) is normal. Therefore \(A_\chi\) has zero normal–tangential components there and normal eigenvalue \(\chi\). Its boundary equation solves for \[ \partial_\nu Z=\mathcal H_s(x,Z,f,Df), \tag{138}\] where \(\mathcal H_s\) is smooth on a fixed bounded variable range. Its derivatives with respect to \(x\) and its displayed scalar arguments are bounded at fixed parameters. The outer condition remains constant Dirichlet. These are different, disjoint smooth boundary components; there is no change of boundary condition along an edge or a corner.

Fourth, absorb the quadratic growth and control \(DZ\). Continuity of \(Z\) supplies the small factor needed for this step. Fix \(p>3\). The linear local \(W^{2,p}\) estimate for the leading nondivergence operator with Equation (138) or outer Dirichlet data has the form, on fixed nested patches \(U_x\Subset V_x\), \[ \|Z\|_{W^{2,p}(U_x)} \le C\left(1+\|DZ\|_{L^{2p}(V_x)}^2 +\|Z\|_{W^{1,p}(V_x)}\right). \tag{139}\] Here the boundary norm is controlled by \[ \|\mathcal H_s(x,Z,f,Df)\|_{W^{1-1/p,p}(\partial V_x)} \le C\bigl(1+\|Z\|_{W^{1,p}(V_x^+)}\bigr), \tag{140}\] using trace and composition after extending its smooth expression through the collar. The derivatives of \(f,Df\) entering this extension are already bounded. The leading matrix has a fixed Hölder modulus, so its small oscillation on sufficiently small patches permits the usual freezing and absorption in the linear estimate. The normal boundary operator is uniformly oblique and satisfies the complementing condition. The general boundary theory is developed in the two papers of Agmon, Douglis and Nirenberg (Agmon et al. 1959, 1964). The scalar estimate used here is (Agmon et al. 1959, Theorem 15.2, estimate (15.5), p. 704).

The outer patches \(U_x,V_x,V_x^+\) and the constant in (139) are fixed before the next radius \(h\) is chosen; only the interpolation is localized further. Let \([Z]_{C^{0,\theta}}\le H_0\). On balls or half-balls of radius \(h\), the scaled Gagliardo–Nirenberg inequality, applied after subtracting a constant, gives \[ \begin{split} \|DZ\|_{L^{2p}(B_h)}^2 &\le C\operatorname{osc}_{B_h}Z\, \|D^2Z\|_{L^p(B_h)}\\ &\quad+Ch^{3/p-2}(\operatorname{osc}_{B_h}Z)^2. \end{split} \tag{141}\] One can use a bounded extension operator on a half-ball; its constants are uniform in the boundary charts. Here the interpolation exponents are spatial dimension \(3\), derivative orders \(2\) and \(1\), top-derivative exponent \(p>3\), height exponent \(\infty\), and interpolation parameter \(1/2\); the resulting gradient exponent is \(2p\). This is (Nirenberg 1959, Lecture II, Equation (2.2), p. 125, and Remark 5, p. 126), including the lower-order term on a bounded patch. Subtracting a constant gives the oscillation, rather than the absolute height, as the coefficient of \(\|D^2Z\|_p\). Cover \(V_x\) by radius-\(h\) patches of bounded overlap and take the \(\ell^p\) sum of Equation (141). It follows that \[ \|DZ\|_{L^{2p}(V_x)}^2 \le CH_0h^\theta\|D^2Z\|_{L^p(V_x^+)} +CH_0^2h^{2\theta-2}. \tag{142}\] Also, for any \(\eta>0\), \(\|Z\|_{W^{1,p}(V_x)} \le\eta\|D^2Z\|_{L^p(V_x^+)}+C_\eta\). Each enlarged patch \(V_x^+\) is covered by a bounded number of the patches \(U_y\), independently of \(R\). Set \(S=\sup_y\|Z\|_{W^{2,p}(U_y)}\), finite for each smooth finite-domain solution. Equations (139) and (142) give \[ S\le C(H_0h^\theta+\eta)S+C(h,\eta). \tag{143}\] Choose \(h\) and then \(\eta\) so the first coefficient is less than \(1/2\). We obtain a uniform \(W^{2,p}\) bound and hence a \(C^{1,1-3/p}\) bound for \(Z\).

Finally, obtain all remaining derivatives. The bounds \(f\in C^{2,\theta}\) and \(Z\in C^{1,1-3/p}\) make the expanded \(Z\) source Hölder and its normal datum \(\mathcal H_s(x,Z,f,Df)\) of class \(C^{1,\theta'}\), after reducing the positive exponent. Its principal coefficients are Hölder as well, so both the interior and normal-boundary Schauder estimates apply. First the nondivergence \(Z\)-equation has Hölder source, giving \(Z\in C^{2,\theta'}\); the scalar \(f\)-equation then gains another derivative. Repetition gives every finite classical derivative supported by the smooth data. The same procedure holds on the uniform unit patches of the asymptotic end and on the large outer spheres, whose normal-coordinate constants are uniform. This proves the asserted independence of \(R\). The prepared background has end symbol bounds of every order by Definition 2; no regularity constant here requires additional derivatives of the original weak-decay data. ◻

Proof. For the scalar continuation below, use the affine space of \(C^{3,b}\) functions with the assigned Dirichlet values and the equation space \(C^{1,b}\). Its individual solutions meet the \(C^3\) hypothesis of Lemma 27; no uniform third-derivative bound is assumed before that lemma is applied. The monotonicity \(\partial Z/\partial t=1+pv/4>0\) defines \(t=t(x,Z,Df)\) smoothly for every finite input, including \(Df=0\). The principal matrix has no dependence on the value of \(f\). The right side is increasing in \(h=\tau f\), with derivative at least one. At a height extremum the collar terms vanish and \(Df=0\), so the equation gives fixed constant upper and lower barriers. Including all prescribed boundary heights, we obtain \[ |h|\le C_0, \tag{145}\] where the constant can be chosen independently of \(Z\) and the homotopy parameter. We include the fixed boundary values in its choice.

Large slopes. Only \(|Z|\le M\) is assumed here. As \(\sigma\to\infty\), the implicit equation forces \(t\to-\infty\) uniformly in \(Z\). There \(p=N\) and \(l=e^{Nt}\), so its exact form is \[ e^{8Z}=e^{8t}+e^{(2N+8)t}\sigma^2. \tag{146}\] Consequently \[ \begin{split} t&=-\frac{\log\sigma}{N+4}+O(1),\qquad l\asymp\sigma^{-N/(N+4)},\\ d&\asymp\sigma^{-8/(N+4)},\qquad \chi\asymp\sigma^{-8/(N+4)},\qquad \frac{\sqrt D}{l}\asymp\sigma. \end{split} \tag{147}\] The constants are uniform for \(|Z|\le M\) at fixed \(N\). Since \(N\ge4\), the exponent \(8/(N+4)\) is at most one. Combining the large-slope estimates with compactness of the remaining variable range yields \[ \chi\ge\frac{c}{1+\sigma},\qquad |G_f|\le C(1+\sigma) \tag{148}\] whenever Equation (145) holds.

Boundary slopes. The estimate \(\chi\ge c/(1+\sigma)\) suggests a concave logarithmic barrier: its axial second derivative can dominate terms of linear growth in its slope. Let \(\delta\) be smaller than a fixed collar radius, half the distance between distinct boundary components, and a normal injectivity radius of a smooth extension of the finite domain. On a boundary distance collar use \[ \psi(r)=k^{-1}\log(1+Br),\qquad \psi''=-k(\psi')^2. \tag{149}\] The upper and lower barriers are the assigned face value plus or minus \(\psi\). On these tests the axis is normal to the distance leaves, so their principal trace is \(\chi\psi''+\psi'\Delta r\). At slopes at least one, Equation (148) gives \[ -k\chi(\psi')^2\le-\tfrac12ck\psi'. \tag{150}\] Choose \(k\) large enough to dominate all the bounded geometry terms and the linear growth in Equation (148). Next shrink \(\delta\) so that \(1/(2k\delta)\) exceeds any fixed large-slope threshold used here. Finally choose \(B\) large enough that \(B\delta\ge1\) and \(\psi(\delta)>4C_0/\tau\). The latter exceeds twice the bound for \(\operatorname{osc}f\) from (145). Then \(\psi'\ge1/(2k\delta)\) on this collar and the two barriers compare by the height monotonicity. This gives a uniform boundary gradient bound.

Interior slopes. Use the same modulus for differences at two points. After increasing \(B\) if needed, consider \[ f(x)-f(y)-\psi(\operatorname{dist}(x,y)), \qquad 0<\operatorname{dist}(x,y)<\delta. \tag{151}\] The distance is taken in the fixed smooth extension. The boundary barriers exclude a positive maximum with one endpoint on a face: the other point lies in that face’s collar, its distance to the face is at most the distance between the pair, and \(\psi\) is increasing. The separation choice prevents endpoints on distinct faces. The inequality \(\psi(\delta)>2\operatorname{osc}f\) excludes the other boundary of the two-point domain. At a positive interior maximum let \(r=\operatorname{dist}(x,y)\) and \(q=\psi'(r)\). The two gradients have length \(q\) and are parallel transports along the connecting geodesic. Vary both endpoints simultaneously in parallel transverse directions. The second variation formula for the short geodesic yields \[ \mathop{\mathrm{tr}}_{e_x^\perp}\mathop{\mathrm{Hess}}f(x) -\mathop{\mathrm{tr}}_{e_y^\perp}\mathop{\mathrm{Hess}}f(y)\le Cqr. \tag{152}\] Varying each endpoint separately along that geodesic gives \(f_{;e_xe_x}(x)\le\psi''(r)\) and \(f_{;e_ye_y}(y)\ge-\psi''(r)\). Subtracting the two scalar equations therefore gives \[ -2C(1+q) \le G_f(x)-G_f(y) \le Cqr+(\chi_x+\chi_y)\psi''(r). \tag{153}\] By Equations (148) and (149), the last term is at most \(-ckq\) at the large comparison slopes. Our choice of \(k\) gives a contradiction. Thus Equation (151) is nonpositive. Reversing \(x,y\) proves the same bound for the absolute difference and, on letting \(y\to x\), a global Lipschitz bound for \(f\). The endpoint variations are interior variations in the smooth extension; the connecting short geodesic need not stay inside \(\Omega_R\). This argument used no continuity modulus for \(\chi_x-\chi_y\); only the common transverse eigenvalues and the radial lower bound were needed.

Continuation and uniqueness. The height and gradient estimates supply compactness for a scalar continuation. Interpolate the normalized equation with \(\Delta f=\tau f\), retaining its prescribed boundary values. Explicitly, for \(a\in[0,1]\) take principal matrix \((1-a)I+aA_\chi\) and right side \((1-a)\tau f+aG_f\). The height sign, the linear gradient growth, and Equation (148) persist uniformly in \(a\). The height and logarithmic gradient bounds just proved are therefore uniform. After these bounds the interpolating equations are uniformly elliptic. Their matrices still have the axial form, with axial eigenvalue \((1-a)+a\chi\), so Lemma 27 applies. Schauder estimates then give a uniform \(C^{2,\theta}\) bound at fixed input \(Z\). The coefficients and right side are now Lipschitz in position on bounded input sets, which permits their \(C^{0,b}\) estimates and an upgrade to \(C^{2,b}\). Differentiating once, or using the next Schauder estimate, yields \(f\in C^{3,b}\) because \(Z\in C^{1,b}\).

The scalar linearization has the form \(a^{ij}D_{ij}\varphi+b^iD_i\varphi-c\varphi\) with \(c\ge c_1\tau>0\) on the bounded variable range. Indeed the principal matrix is independent of the height, while \(\partial_fG_f\ge\tau\sqrt D/l>0\). The Dirichlet maximum principle gives uniqueness and a zero kernel; the linear elliptic Dirichlet theory gives an inverse from \(C^{1,b}\) to \(C^{3,b}_0\), where the subscript denotes zero data on the entire boundary. The implicit-function theorem gives openness of the interpolation parameter set. The uniform classical bounds and compactness give closedness, starting from the invertible Laplace endpoint. This proves existence. The same maximum principle applied to the difference of two solutions proves uniqueness of the nonlinear solution. The implicit-function theorem, or compactness and uniqueness, gives continuous dependence on \(Z\) and on the homotopy parameter. At the finitely many concatenation points the scalar equations agree, so the same continuity holds there. ◻

Proof. The fixed points in \(\overline{\mathcal U}\) form a compact set \(\mathfrak F\). Indeed, boundedness of \(\mathcal U\), compactness of \(I\) and of the images of \(T\) give a convergent subsequence of any sequence of such fixed points; continuity preserves the fixed-point equation. The hypothesis places \(\mathfrak F\) inside \(\mathcal U\). Also \(\{s\}\times\partial\mathcal U_s\) lies in the relative boundary of \(\mathcal U\), so each section degree is defined.

Fix \(s_0\in I\). Cover its compact fixed-point slice by finitely many open balls in \(X\) whose closed balls remain in \(\mathcal U_s\) for all \(s\) sufficiently close to \(s_0\). Relative openness and the finite cover permit this common parameter interval. Let \(U\) be their union. All fixed points in the closed sections for these nearby parameters lie in \(U\) after shortening the interval: otherwise compactness of \(\mathfrak F\) would give an omitted fixed point in the slice at \(s_0\). Excision identifies the degree on each \(\mathcal U_s\) with that on \(U\), and fixed-domain homotopy invariance makes the latter constant. If the slice is empty, compactness instead gives no nearby fixed points in the closed sections, so those degrees are all zero. Thus the section degree is locally constant, and connectedness of \(I\) makes it constant. These are the excision and homotopy properties of Leray–Schauder degree (Leray and Schauder 1934, secs. 12–14, pp. 59–61); see also (Deimling 1985). At the stated terminal map, normalization gives degree one; the existence property of nonzero degree proves the last assertion. ◻

Proof. Choose the parameters before defining degree. Fix the prepared data, thresholds, collars and \(\epsilon\). Choose an integer \(N_0\) so that \[ N_0\ge\max\{4,\lfloor2/\epsilon\rfloor+1\} \tag{154}\] and so that the floor test and all estimates of Proposition 26 apply for \(N\ge N_0\). Let \(R_{\mathrm{ap}}(N)\) dominate every lower-radius requirement in the a priori estimates and in the floor exclusion; let \(R_{\mathrm{geo}}\) contain all compact boundary faces, collars and supports of the construction. The end barrier of that proposition gives a constant \(C_{\mathrm{out}}\), independent of \(N,R,s\), such that every admissible or floor-contact homotopy solution has \[ |\nabla f|\le C_{\mathrm{out}}\tau^{-1}R^{-1-\gamma} \quad\text{on the outer sphere}. \tag{155}\] The scalar last stage has \(f=0\) and satisfies the same bound. Choose, enlarging to a radius of a smooth truncation if necessary, \[ \begin{split} R_{\min}(N)=\max\biggl\{&N,R_{\mathrm{geo}},R_{\mathrm{ap}}(N),\\ &\left(\frac{2C_{\mathrm{out}}\ell} {\tau\sqrt{e^{8\epsilon}-1}} \right)^{1/(1+\gamma)}\biggr\}. \end{split} \tag{156}\] At \(Z=0\), the value of its implicit expression at \(t=-\epsilon\) is \(-\epsilon+\frac18\log(1+\ell^2\sigma^2)\). Equations (155) and (156) make this strictly negative. Strict monotonicity of that expression in \(t\) therefore excludes an outer floor contact. The term \(N\) in Equation (156) ensures \(R_{\min}(N)\to\infty\).

The sublevel and collar estimates apply to arbitrary smooth admissible or floor-contact solution sequences with \(N\to\infty\) and expanding outer cutoffs, uniformly over the compact ranges of the homotopy parameters. If the needed conclusion failed for arbitrarily large \(N\) with \(R\ge R_{\min}(N)\), one could select such a violating sequence. The chosen radius convention would make it precisely an expanding sequence excluded by those proofs. Thus a single \(N_0\), followed by the radius choices above, suffices for every homotopy parameter. No upper bound for \(Z\) is inserted in the earlier contact estimates: there \(\sigma\asymp\tau^{-1}\) and \(l\ge\ell\) alone give \(d\lesssim(\tau/\ell)^2\).

Define the compact map on all bounded input sets. For the rest of the degree argument fix \(N\ge N_0\) and \(R\ge R_{\min}(N)\). Work in the Banach space \[ X_b=\{Z\in C^{1,b}(\overline{\Omega_R}): Z=0\text{ on the outer face}\}, \qquad 0<b<1. \tag{157}\] For an input \(Z\in X_b\), first solve \(f=\mathcal S_s(Z)\) by Lemma 30. Compute \(t,u,A_\chi,\mathcal T\) and the full homotopy source from this input pair. Write \(B_s\) for the frozen drift summand in the flux, \(E_s\) for the full right side of its divergence equation, and \(q_s\) for the prescribed inner flux. Thus before the final scaling \(B_s=\lambda_suA_\chi K(w,\cdot)\) and \(E_s=u\Xi\); on the final interval \(E_s\) and \(q_s\) have the simultaneous scalar factor prescribed in Proposition 15. Define \(\mathcal K_s(Z)=\widetilde Z\) by the linear mixed problem \[ \begin{cases} \mathop{\mathrm{div}}(4uA_\chi\nabla\widetilde Z+B_s)=E_s &\text{in }\Omega_R,\\ (4uA_\chi\nabla\widetilde Z+B_s)_\nu=q_s &\text{on }B,\\ \widetilde Z=0 &\text{on the outer face}. \end{cases} \tag{158}\] All coefficients in this problem are computed from the input \((\mathcal S_s(Z),Z)\); the only unknown is \(\widetilde Z\). The normal \(\nu\) on the inner faces points into the domain. For a test function \(\varphi\) vanishing on \(S_R\), the weak form is \[\int_{\Omega_R}4uA_\chi\nabla\widetilde Z\cdot\nabla\varphi =-\int_{\Omega_R}E_s\varphi -\int_{\Omega_R}B_s\cdot\nabla\varphi -\int_B q_s\varphi.\] The sign of the boundary term comes from using \(-\nu\) as the integration normal. The scalar estimate bounds the leading coefficient above and away from zero on each bounded input set. The nonempty outer Dirichlet face supplies Poincaré’s inequality, so the bilinear form is coercive and has a unique solution.

On bounded input sets in \(C^{1,b}\), Lemma 30 bounds \(f\) in \(C^{3,b}\). Hence the leading coefficient and drift in Equation (158) are \(C^{1,b}\), the bulk source is \(C^{0,b}\), and the prescribed boundary data are \(C^{1,b}\). The derivatives of \(t(x,Z,Df)\) involve only \(DZ\) and \(D^2f\), so this assertion includes the quadratic expression \(\mathcal T\). Linear regularity gives a bounded output set in \(C^{2,b}\). The inclusion \(C^{2,b}\hookrightarrow C^{1,b}\) is compact on the finite domain. Continuous dependence of the scalar solve and the linear problem makes \((s,Z)\mapsto\mathcal K_s(Z)\) a continuous compact homotopy on bounded sets. These assertions hold also at the concatenation parameters because the defining equations coincide there. A fixed point solves the coupled system; repeated Schauder estimates make it smooth.

Exclude fixed points from the boundary of admissibility. Write \(t_s[Z]=t(x,Z,D\mathcal S_s(Z))\). The scalar solution must be computed before this admissibility test. Define \[ \mathcal O=\{(s,Z)\in[0,4]\times X_b: \min_{\overline{\Omega_R}}t_s[Z]>-\epsilon, \ \|Z\|_{C^{1,b}}<M_1\}. \tag{159}\] It is relatively open because the scalar solve and the implicit function defining \(t\) depend continuously on \((s,Z)\). Propositions 26 and 29 give a uniform bound in \(C^{1,b}\) for all fixed points satisfying \(t\ge-\epsilon\). If initially needed, use their higher classical bounds to reach the chosen exponent \(b\). Choose \(M_1\) larger than this bound. No fixed point lies on the norm boundary of \(\mathcal O\). No fixed point lies on its floor boundary either: outer contacts were excluded above and all remaining contacts are excluded by Proposition 20.

Apply degree on the varying admissible sections. The sections \(\mathcal O_s\) need not be constant in \(s\). Lemma 31 applies with \(I=[0,4]\), \(X=X_b\), \(T_s=\mathcal K_s\) and \(\mathcal U=\mathcal O\). We have proved continuity and compactness on bounded input sets; the norm condition makes \(\mathcal O\) bounded and the scalar solve makes it relatively open. Its relative boundary can contain only norm contact or floor contact, and neither can be a fixed point by the preceding estimates. Thus its section degree is constant.

At \(s=4\), the scalar equation has the unique solution \(f=0\). The drift vanishes, and both \(E_4\) and \(q_4\) vanish for every input. Equation (158) then has output identically zero. Moreover, \(t_4[0]=0>-\epsilon\), so \(0\in\mathcal O_4\) and \[ \deg(I-\mathcal K_4,\mathcal O_4,0)=1. \tag{160}\] Homotopy invariance gives the same degree at \(s=0\). There is therefore a fixed point in \(\mathcal O_0\), which is the desired smooth solution with \(t>-\epsilon\).

Remove only the outer truncation. Keep \(N\) and all other deformation parameters fixed. Proposition 29 gives estimates independent of \(R\) on an exhaustion by compact sets, using boundary charts at the fixed inner faces. A diagonal subsequence converges smoothly on compact subsets of \(\overline\Omega\). Both equations and their inner boundary conditions pass to the limit. The strict finite-domain inequality gives the non-strict limit inequality \(t\ge-\epsilon\). The following section establishes end normalization, decay and mass flux; those conclusions do not follow from compact exhaustion alone. ◻

Proof. The compact convergence, including at the fixed inner boundary, follows from Theorem 32 and its uniform fixed-\(N\) classical estimates. All equations and inner boundary conditions pass to this limit. To preserve the end normalization, we prove the decay estimates on the truncations before passing to infinity.

Outside a fixed large sphere, \(K=C=0\), and the first equation is \[\mathop{\mathrm{tr}}_{A_\chi}\mathop{\mathrm{Hess}}_g f=\frac{\tau\sqrt D}{l}\,f.\] At fixed \(N\) the preceding estimates make the left side uniformly elliptic, and its positive height coefficient is bounded below by a positive constant depending on \(N\). It is bounded above as well. For sufficiently small \(c_N>0\), the functions \(\exp[-c_N(r-r_0)]\) are positive supersolutions of the corresponding linear equation outside a sufficiently large \(r_0\): their radial second derivative has size \(c_N^2\), while their tangential second derivatives have size \(c_N/r\), and the fixed positive height term dominates both. Multiplying by a constant bounds the solution on \(S_{r_0}\); the zero outer Dirichlet value is also bounded. The maximum principle applied to both signs of \(f\) gives the uniform exponential estimate. The fixed-\(N\) unit-patch estimates, followed by the local homogeneous linear estimates for this equation, give the same decay for derivatives. The corresponding boundary estimates apply on the large outer spheres. Iterating the smooth equations gives each required finite derivative order.

The relation \(Z=t+\tfrac18\log(1+l^2|\nabla f|^2)\) now shows that \(t-Z\) is exponentially small, with derivatives. The graph errors have the same property. On the end the second equation becomes \[ \Delta_gZ+b_N(Z)|dZ|_g^2=O(r^{-4}),\qquad b_N(s)=p_L(s)-\delta_0\bigl(p_L(s)-1\bigr). \tag{162}\] Indeed \(A_\chi=I\) and \(u=L(Z)\) up to exponentially small errors, whereas \(\mathcal T=4(p_L(Z)-1)|dZ|^2\) up to such errors. The \(\rho_0\) penalty weight is \(O(r^{-4})\); the term with \(\rho_1\) is multiplied by the exponentially small \(v\). All coefficients in this statement are bounded on the fixed-\(N\) solution range. Differentiably controlled exponential errors can absorb any fixed polynomial power of \(r\).

Define an increasing smooth function on that range by \[\Phi_N(0)=0,\qquad \Phi_N'(s)=\exp\!\left(\int_0^s b_N(z)\,\mathrm dz\right).\] Its derivative is bounded above and below by positive fixed-\(N\) constants. The chain rule in (162) gives \[\Delta_g\Phi_N(Z)=O(r^{-4}).\] The function is bounded and vanishes at the outer cutoff. To obtain a bound independent of that cutoff, choose \(0<\eta<1\). For large \(r\), a positive multiple of \(r^{-1}-r^{-1-\eta}\) satisfies \[-\Delta_g(r^{-1}-r^{-1-\eta})\ge c r^{-3-\eta}.\] Here the Euclidean leading term is \(\eta(1+\eta)r^{-3-\eta}\), and the metric error is \(O(r^{-4})\). It dominates the right side after increasing the multiple and the fixed inner radius. Comparison for both signs yields \(|\Phi_N(Z)|\le C'(N) r^{-1}\), where \(C'(N)\) is a fixed-\(N\) constant with no polynomial claim. Consequently \(Z=O(r^{-1})\).

At sufficiently large radius this estimate and the exponential graph error give \(t>-\epsilon+1/N\); hence the floor penalty vanishes there. Rescale each annulus of radii comparable to \(r\) to unit size. The equation for \(\Phi_N(Z)\) has a bounded rescaled source and uniformly controlled metric coefficients. Local \(W^{2,p}\) estimates with \(p>3\) first give the gradient decay. Its source is then a smooth weight times a smooth function of the bounded variables, plus the already controlled exponential errors. Schauder estimates, or a further Sobolev bootstrap followed by them, give \(Z=O_2(r^{-1})\). Inverting \(\Phi_N\) preserves these bounds, and the same holds for \(t\).

Since \(L(0)=1\), we have \(u=1+O(r^{-1})\) and \(A_\chi=I+O(e^{-c_N r})\). With \(K=0\) on the end, the definition of \(V\) gives (161). Moreover \(\mathop{\mathrm{div}}V=u\Xi\) is integrable there: \(\mathcal T=O(r^{-4})\), \(\rho=O(r^{-4})\), the penalty has vanished, and \(\tau a\sigma\) decays exponentially. The divergence theorem on annuli proves the existence of the outward flux limit. ◻

Proof. At the physical endpoint the curvature identity gives \[\begin{align*} \tfrac12e^{4t}R_{\hat g} ={}&8\pi(\mu+J(w))+(1-\delta_0)\mathcal T +(1-\delta_0)\tau a\sigma\\ &+w\cdot\nabla C-F\mathop{\mathrm{tr}}K-\rho+m_0(t)\mathcal P. \end{align*}\] The height bounds on the compact supports of \(K\) and \(C\) give \(h\in[b_-,b_+]\). Since \(|w|\le1\), the terms \(|F\mathop{\mathrm{tr}}K|+|\nabla C|+\rho\) are dominated by \(8\pi(\mu-|J|)\), by the choices in Proposition 11 and the choice of \(\rho\). Outside those supports the same conclusion uses \(8\pi(\mu-|J|)-\rho\ge0\). All remaining terms are nonnegative. This proves \(R_{\hat g}\ge0\).

The boundary identity and the prescribed flux give \(H+4\partial_\nu t=-N_B<0\). Since \(f\) is constant on each face, this is exactly the mean-curvature sign for \(\hat g\). Also \(\hat g\ge e^{-4\epsilon}g\), so any escaping curve has infinite length; the compact boundary is included. Smoothness and completeness therefore follow.

Lemma 33 gives the stated metric decay. The prepared-data hypothesis supplies \(R_g=O(r^{-3-\delta})\) on the end. Since graph errors are exponentially small, the conformal formula gives \[R_{\hat g} =e^{-4t}\bigl(R_g-8\Delta_gt-8|dt|_g^2\bigr) +O(e^{-c_N r}).\] Equation (162), the vanishing far-end penalty, and the gradient estimate give \(\Delta_gt=O(r^{-4})\). This proves the asserted scalar-curvature decay, including the pointwise falloff required by the AF convention of (Bray 2001, Definition 21 and Theorem 19).

Exponential graph errors contribute no ADM flux. The conformal change and \(t=O_2(r^{-1})\) give \[E_{\hat g} =E-\frac1{2\pi}\lim_{r\to\infty} \int_{S_r}\partial_\nu t\,\mathrm dA_g.\] Replacing the metric normal and area form by their Euclidean versions costs \(o(1)\), and the \(O(r^{-3})\) error in (161) has vanishing sphere integral. This proves (163). ◻

Proof. The normal \(\nu\) on \(B\) points into \(\Omega\), so the divergence theorem has the sign \[ \mathfrak F_N =\int_\Omega u\Xi\,\mathrm dV_g+\int_BV_\nu\,\mathrm dA_g. \tag{164}\] Each integral is finite for a fixed \(N\), by Lemma 33 and compact smoothness.

At a black face, \(F-P_B=H\) and \(w_\nu=a\). At a white face, \(F-P_B=-H\) and \(w_\nu=-a\). Thus at either type of face, \[V_\nu/u=-(1-a)H-N_Bd\ge-Cd.\] The signed boundary gradient bounds give \(\sigma\ge c/\tau\), so \[\sqrt d\le\frac1{l\sigma}\le C\tau/\ell,\qquad \int_BV_\nu\,\mathrm dA_g\ge -C\frac{\tau}{\ell}\int_BL\,\mathrm dA_g.\] Use the polynomial collar trace estimate from Section 5 with the constant test function: \[\int_B L\,\mathrm dA_g \le C(1+N)^C \int_{\mathcal C}u(1+\sqrt{\mathcal T})\,\mathrm dV_g,\] where \(\mathcal C\) is a fixed compact union of collars. Its polynomial constant is independent of the upper \(Z\) bound. Since \(\rho\) has a fixed positive minimum on \(\mathcal C\), \[\int_{\mathcal C}u(1+\sqrt{\mathcal T})\,\mathrm dV_g \le C\int_\Omega u(\delta_0\mathcal T+\rho)\,\mathrm dV_g.\] The factor \(C(1+N)^C\tau/\ell =C(1+N)^C\ell^{1/2}\) tends to zero. For large \(N\), the entire negative boundary term in (164) is therefore at most one quarter of \(\int_\Omega u(\delta_0\mathcal T+\rho)\,\mathrm dV_g\).

It remains to estimate the floor penalty. On its support, \(-\epsilon\le t\le-\epsilon+1/N\), so \(l\) and \(L\) are comparable to \(\ell\), with constants independent of \(N\). Directly from their definitions, \[\begin{split} u&=L\sqrt{1+l^2\sigma^2} \le C(\ell+\ell^2\sigma),\\ uv&=\frac{Ll^2\sigma^2}{\sqrt{1+l^2\sigma^2}} \le C\ell^2\sigma,\\ ua\sigma&=Ll\sigma^2\ge c\ell^2\sigma^2 . \end{split}\] Consequently \[um_0\mathcal P \le C_N\bigl[\ell\rho_0+\ell^2\sigma(\rho_0+\rho_1)\bigr].\] Young’s inequality, with the square term chosen to be a quarter of \(\delta_0\tau ua\sigma\), yields \[um_0\mathcal P \le \tfrac14\delta_0\tau ua\sigma+ C_N\left[\ell\rho_0+ \frac{\ell^2}{\tau}(\rho_0^2+\rho_1^2)\right].\] Squaring the constant only changes the polynomial \(C_N\). The integrals of \(\rho_0\), \(\rho_0^2\), and \(\rho_1^2\) are finite: on the end their orders are respectively \(r^{-4}\), \(r^{-8}\), and \(r^{-4\gamma}\), with \(4\gamma>3\). Integrating leaves at most \(C(1+N)^C(\ell+\ell^2/\tau)\), after absorption.

Substitute both bounds in (164). Positive fractions of the three bulk terms \(\delta_0\mathcal T,\rho,\delta_0\tau a\sigma\) remain. Dropping them proves the claim. Finally \(\ell^2/\tau=\ell^{1/2}\), so the error tends to zero exponentially against every polynomial loss. ◻

Proof of Theorem 3. The fixed subexterior and preservation of every full cut are supplied by Proposition 11. Fix \(\epsilon\) and take the global solutions of Lemma 33, obtained by first exhausting the end at fixed \(N\). Lemma 34 gives completeness, the two curvature signs, and the end decay. The pointwise inequality \(\widehat g\ge e^{-4\epsilon}g\) gives the two-dimensional area inequality on every full cut, including every component and every coincident portion. The energy formula and Lemma 35 give the asserted upper error with \(\eta_{\epsilon,N}=\varepsilon_N/(8\pi)\). This completes the geometric construction before any Riemannian Penrose comparison. ◻

Proof. For each fixed sufficiently large \(N\), minimize \(\hat g\)-area among filled cuts enclosing \(B\), using a large outer sphere as an auxiliary barrier. The independent enclosure results of (OpenAI 2026a, Lemmas 2.1–2.3) apply to this smooth AF metric; we check their hypotheses and the strict detachment here. A fixed enclosing competitor bounds the area, and local perimeter density estimates prevent minimizers from escaping to infinity. The positive mean curvature of sufficiently large outer spheres excludes contact there, as in that enclosure theorem. For the inner barrier, let \(U_s\) be a short outward normal collar of the filled obstacle and \(Y\) its outward unit foliation field. Strict negativity gives \(\mathop{\mathrm{div}}_{\hat g}Y<0\) throughout the collar. If a perimeter minimizer \(E\) omitted a positive-volume portion \(D=U_s\setminus E\), Gauss–Green and \(|Y|_{\hat g}\le1\) would give \[P_{\hat g}(E\cup U_s)-P_{\hat g}(E) \le\int_D\mathop{\mathrm{div}}_{\hat g}Y\,\mathrm dV_{\hat g}<0.\] Thus \(E\) contains a whole inner collar; obstacle regularity upgrades this inclusion to its regular representative. Its free boundary is disjoint from the obstacle. The resulting cut \(\Gamma_N\) is compact, smooth, and minimal in dimension three; it may be disconnected. It is outer area-minimizing by its minimizing property. Equivalently one may first use the ordinary trapped-region theorem for \((\hat g,0)\) and then take the outer minimizing enclosure. The boundary version of the Riemannian Penrose theorem applies to the exterior of \(\Gamma_N\), by Lemma 34 and (Bray 2001, Theorem 19).

For every two-plane the restriction of \(\bar g=g+l^2df^2\) dominates that of \(g\). The floor therefore gives \[|\Gamma_N|_{\hat g} \ge e^{-4\epsilon}|\Gamma_N|_g \ge e^{-4\epsilon}A_*.\] The last inequality uses that \(B\), and hence \(\Gamma_N\), cuts off the original filled obstacle. Lemmas 34 and 35 now give \[E+\frac{\varepsilon_N}{8\pi} \ge E_{\hat g} \ge\sqrt{\frac{|\Gamma_N|_{\hat g}}{16\pi}} \ge e^{-2\epsilon}\sqrt{\frac{A_*}{16\pi}}.\] Let \(N\to\infty\) with \(\epsilon\) fixed, and then \(\epsilon\downarrow0\). This proves the reduced inequality.

Finally apply the weak three-dimensional rest-preparation theorem of (OpenAI 2026b, Proposition 7.13) to an exterior satisfying the hypotheses of Theorem 4. Write \((g_j,K_j)\) for the resulting data. Their tensor fields have compact support, their momenta vanish, and their metrics have all-order \(r^{-1}\) decay and the stipulated scalar falloff. Discard a finite initial segment so that \(E_j>0\). Strict trapping and strict DEC hold on the compact core. The positive minimum there of \(8\pi(\mu_j-|J_j|_{g_j})\varrho^{3+\delta}\), together with the end margin, gives the global weighted bound after decreasing \(c_j\). The full enclosing area is positive by the geometric enclosure lemma. Thus each fixed \(j\) satisfies Definition 2; no constant uniform in \(j\) has been used in its deformation solve. The reduced energies converge to its invariant ADM mass and the full enclosing infima converge to the original enclosing infimum. Applying the reduced bound to each member of that sequence and passing to the limit gives Equation (10). ◻

Proof. Choose a positive smooth function \(b\ge|K|_g\) which equals a constant multiple of \(r^{-1-q}\) sufficiently far out. We first solve \[ -\Delta_g\phi =b\sqrt{|\,\mathrm d\phi|_g^2+r^{-6}}+r^{-4-\delta}, \qquad \partial_\nu\phi=-1\text{ on }S, \qquad \phi\longrightarrow0\text{ at infinity}, \tag{183}\] where \(\nu\) points into the exterior. In particular, the normal derivative in this equation is the negative of the domain’s outward normal derivative at its inner boundary.

Here are the existence and decay details. On the compact truncation \(M_R\), impose \(\phi=0\) on the outer coordinate sphere and replace \(b\) on the right by \(\lambda b\), \(0\le\lambda\le1\). The linearization, with homogeneous boundary data, is \[v\longmapsto-\Delta_gv -\lambda b\frac{\langle\,\mathrm d\phi,\,\mathrm dv\rangle_g} {\sqrt{|\,\mathrm d\phi|_g^2+r^{-6}}}.\] Its drift is bounded by \(b\). Its kernel is zero by the strong maximum principle and the boundary point lemma, since the outer Dirichlet face is nonempty and the inner face has homogeneous normal derivative. The mixed operator has Fredholm index zero, by deformation to the mixed Laplacian, and hence is invertible. The two boundary components are disjoint, smooth, and compact, so ordinary scalar elliptic estimates apply without a junction between boundary types; see (Gilbarg and Trudinger 2001, chaps. 3, 6, and 9) and (Agmon et al. 1959, Theorems 7.3 and 15.2–15.3).

For fixed \(R\), the global mixed \(W^{2,p}\) estimate, \(p>4\), and interpolation give \[ \|\phi\|_{W^{2,p}(M_R)} \le C_{R,p}\bigl(1+\|\,\mathrm d\phi\|_{L^p(M_R)} +\|\phi\|_{L^p(M_R)}\bigr) \le C'_{R,p}(1+\|\phi\|_\infty). \tag{184}\] These constants are uniform in \(\lambda\). If the suprema were unbounded, divide by their sup norms and take a subsequence. Compact embedding of \(W^{2,p}\) in \(C^{1,\alpha}\), with a smaller \(\alpha>0\), gives a nonzero limit \(v\) satisfying \[-\Delta_gv=\lambda_*b|\,\mathrm dv|_g, \qquad \partial_\nu v=0\text{ on }S, \qquad v=0\text{ on the outer face}.\] This is a homogeneous linear equation with a bounded measurable drift: set the drift to \(\lambda_*b\mathop{\mathrm{grad}}v/|\,\mathrm dv|\) where \(\,\mathrm dv\ne0\), and to zero otherwise. The maximum principle and boundary point lemma again exclude a nonzero solution. Thus (184) is bounded, and Schauder estimates and bootstrapping give closedness in the continuity argument. The \(\lambda=0\) mixed Poisson problem is solvable, so the implicit function theorem and compactness give a smooth solution for \(\lambda=1\). It is nonnegative: a negative minimum is excluded in the interior by \(-\Delta\phi>0\), on the outer face by its zero value, and on the inner face by \(\partial_\nu\phi=-1\).

We next bound these solutions independently of \(R\). On the end consider \[T(r)=r^{-2}(1-Ar^{-\delta}).\] Choose a fixed \(A>0\), then a large fixed \(r_0\), so that \(T>0\) and \[ (-\Delta_g-b|\,\mathrm d\,\cdot\,|)T \ge c\,r^{-4-\delta}\qquad(r\ge r_0). \tag{185}\] Indeed, \(-\Delta_\mathbb RT=A\delta(2+\delta)r^{-4-\delta}\), while both the metric error and \(b|\,\mathrm dT|\) are \(O(r^{-4-q})\). Also \[-\Delta_g\phi-b|\,\mathrm d\phi| \le b r^{-3}+r^{-4-\delta}\le C r^{-4-\delta}.\] Comparison on \(\{r_0\le r\le R\}\), including the zero outer value, therefore yields \[ 0\le\phi_R(x)\le C(1+m_R)r(x)^{-2}, \qquad m_R=\max_{\{r\le r_0\}}\phi_R. \tag{186}\] If \(m_R\) were unbounded along expanding truncations, divide by \(m_R\). Local versions of (184), including the inner boundary estimates, give a nonnegative limit attaining value one in the fixed core. It solves \(-\Delta v=b|\,\mathrm dv|\), has homogeneous inner normal derivative, and is \(O(r^{-2})\) by (186). Its positive maximum is therefore attained at a finite point. The strong maximum principle or the boundary point lemma excludes it. We have a uniform core bound, and diagonal compactness gives a smooth solution of (183).

Rescale annuli by their radii. The bound \(\phi=O(r^{-2})\), the scaled metric bounds furnished by \(g-\delta=O_2(r^{-q})\), and the linear gradient growth in (183) first give scaled \(W^{2,p}\) estimates by interpolation. They give Hölder control of the gradient; the right side then has the scaled Hölder control needed for Schauder estimates. Hence \[ \phi=O_2(r^{-2}). \tag{187}\] The source in (183) is \(O(r^{-4-q})+O(r^{-4-\delta})\), and is integrable. The divergence theorem shows that \[I_\phi=\lim_{R\to\infty}\int_{S_R}\partial_{\nu_g}\phi\,\mathrm dA_g\] exists. Replacing this by its Euclidean-coordinate flux changes it by \(O(R^{-q})\), which tends to zero.

For small \(s>0\) set \(g_s=e^{2s\phi}g\) and \(K_s=e^{s\phi}K\). The trace remains zero. The connection change is \[\Gamma(g_s)^k_{ij}-\Gamma(g)^k_{ij} =s(\delta_i^k\phi_j+\delta_j^k\phi_i-g_{ij}\phi^k).\] Contracting it against the scaled trace-free tensor, and using the four-dimensional scalar-curvature formula, gives \[ \begin{split} e^{2s\phi}\mu_s &=\mu-3s\Delta_g\phi-3s^2|\,\mathrm d\phi|^2,\\ e^{s\phi}J_s &=J+3sK(\mathop{\mathrm{grad}}\phi,\cdot),\\ \Theta_{K_s}(S) &=e^{-s\phi}\bigl(\Theta_K(S)+3s\partial_\nu\phi\bigr). \end{split} \tag{188}\] The last coefficient is three because the boundary has dimension three. A covector norm contributes one more factor \(e^{-s\phi}\), so the first two identities and \(\mu=J=0\) imply \[\begin{align*} e^{2s\phi}(\mu_s-|J_s|_{g_s}) &\ge 3s\bigl[-\Delta\phi-|K||\,\mathrm d\phi|-s|\,\mathrm d\phi|^2\bigr] \\ &\ge3s\bigl[r^{-4-\delta}-s|\,\mathrm d\phi|^2\bigr]. \tag{189}\end{align*}\] The ratio \(|\,\mathrm d\phi|^2/r^{-4-\delta}\) is bounded, by (187) and \(\delta<1\). Thus small \(s>0\) gives a positive gap of the asserted order. The boundary expansion is \(-3s e^{-s\phi}<0\). Moreover, \[R_{g_s}=e^{-2s\phi} \bigl(|K|^2-6s\Delta\phi-6s^2|\,\mathrm d\phi|^2\bigr) =O(r^{-4-\delta}),\] because \(2+2q>4+\delta\). The original metric and tensor decay is preserved, since \(q<2\).

Only \(-6s\partial_i\phi\) survives from the conformal change in the ADM integrand. Products with a metric error or with \(\phi\) have vanishing integrated flux by (187). Thus \[ E_{g_s}=E_g-\frac{s}{\omega_3}I_\phi\longrightarrow E_g. \tag{190}\] Since \(\phi\) is bounded, the comparison \(e^{-2s\|\phi\|_\infty}g\le g_s\le e^{2s\|\phi\|_\infty}g\) holds on the entire exterior. Its three-dimensional area comparison holds for every enclosing cut, and therefore for their infima. In particular, \(\mathcal A_{g_s}(S)\to\mathcal A_g(S)\), with no compactness assumption on minimizing cuts.

Proof. The given diffeomorphism of the exterior permits a smooth compact filling behind \(S\). Extend \(g,K\) smoothly over it; the extended metric can be chosen positive definite by interpolation with any interior metric away from a collar. This extension is used only in the present geometric construction.

Fix a large radius \(R_0\) beyond which every coordinate sphere has \(\Theta_K=3/r+O(r^{-1-q})>0\). Outside a still larger fixed radius, smoothly cut the metric to the Euclidean metric and the unshifted tensor to zero. Dilated cutoffs preserve the bounds \(g-\delta=O_2(r^{-q})\), \(K=O_1(r^{-1-q})\), and consequently the positive expansion of all spheres in the cutoff annuli. Denote these complete auxiliary data by \((g^{\rm aux},K^{\rm aux})\). Choose a fixed nonnegative smooth cutoff \(\xi\), equal to one on the filling and throughout a large neighborhood of \(\{r\le R_0\}\), and zero far out. For \(\max_S\Theta_K<b<0\), use the tensor \[K_b^{\rm aux}=K^{\rm aux}-\tfrac b3\xi g^{\rm aux}.\] Where \(\xi=1\) and the data are uncut, the exact identity is \[ \Theta_{K_b^{\rm aux}}(\Sigma)=\Theta_K(\Sigma)-b. \tag{195}\] In the cutoff region the added tangential trace is \(-b\xi\ge0\), so all the distant spheres still have positive expansion, uniformly in \(b\).

A smooth weakly trapped bounded region cannot meet those distant spheres: at a point where its boundary reaches its greatest radius, its tangent plane and outward normal agree with those of the coordinate sphere, and its mean curvature is at least the sphere’s mean curvature. The tensor traces there agree. This contradicts nonpositive expansion. The same argument applies to the outer frontier furnished by the total-region theorem. Thus all total regions are contained in the uncut part of the construction.

For every indicated \(b\), strictness at \(S\) gives a short outward collar whose leaves have expansion less than \(b\). The filling together with this collar is a nonempty trapped region for \(K_b^{\rm aux}\). The theorem therefore gives a total closed region \(\mathcal D_b\) with smooth stable frontier. Its frontier lies strictly outside \(S\), in the original data, and has \(\Theta_K=b\) by (195). We used one filling and one set of cutoffs for the entire interval of \(b\). Since \(\xi\ge0\), increasing \(b\) decreases the shifted expansion on every candidate hypersurface. It follows that \[ b_1<b_2\quad\Longrightarrow\quad \mathcal D_{b_1}\subset\mathcal D_{b_2}. \tag{196}\]

Choose a compact enlargement \(Q\) containing all these regions. Lemma 6 applies to the increasing family of closed subsets \(E_b=\mathcal D_b\) of this fixed compact metric space, with \(\max_S\Theta_K<b<0\). The preceding construction proves both the common compact containment and monotonicity; every member is nonempty. Choose \(c\) outside the lemma’s countable exceptional set. Its conclusion is exactly (193). Filling bounded complementary pockets ensures that the remaining exterior is the connected component containing the sole end. Any cut enclosing its entire boundary \(B\) also encloses \(S\), proving the area assertion.

It remains to construct the collars. Apply Lemma 7 with ambient dimension four, \(Q=K\), threshold \(c\), and each connected component of \(B\). The total-region theorem gives a compact smooth stable frontier; its principal eigenvalue is nonnegative (Andersson et al. 2008, Proposition 5.1). On an open neighborhood of this frontier the auxiliary cutoff \(\xi\) equals one and the data are uncut, so the relevant shifted tensor is exactly \(K-(c/3)g\). Its expansion differs from \(\Theta_K\) by the constant \(c\), with zero derivative under graph variations. Replacing one component by a sufficiently short outward graph of expansion at most \(c\) would strictly enlarge the total region while preserving every other component. Thus the local maximality hypothesis of the collar lemma holds in this application.

For clarity, its zero-mode construction uses the expansion \(\mathcal E(v)\) of the outward graph on this one component and \(L=D\mathcal E(0)\). The common augmented inverse specializes to \[ (v,a)\longmapsto\left(Lv-a,\ \int_Bv\,\mathrm dA_g\right), \qquad C^{2,\alpha}(B)\times\mathbb R \longrightarrow C^{0,\alpha}(B)\times\mathbb R. \tag{197}\] Here the integrals are over the chosen component. The positive principal and adjoint eigenfunctions, their simple kernels, and the positive pairing used to invert this map are those in (Andersson et al. 2008, sec. 4, Lemma 4.1), as in the common proof. That proof gives strictly larger expansion on every positive outward leaf, including the zero-eigenvalue case; stability alone would not give this conclusion. Finitely many components permit disjoint shortened collars, proving (194). ◻

Proof. First, every set under consideration lies inside one fixed large sphere. Otherwise the coordinate sphere at its maximum radius is an exterior support with positive expansion, contrary to \(k'<0\). Choose a compact enlargement containing this sphere and a small metric neighborhood of it. All constants below depend only on the smooth geometry and tensor on this enlargement. Distances may be computed in the complete smooth filling, since the contact points and sufficiently short segments used below lie in the original exterior: \(A\) contains the filling and a collar of \(S\).

For \(z>0\), put \(A_z=\{y:\mathop{\mathrm{dist}}(y,A)\le z\}\). Apply Lemma 9 in ambient dimension four, with its set \(D=A\), tensor \(Q=K\), and threshold \(k'\). The compact neighborhood and short segments required by that local lemma are precisely those just fixed. The definition (198) is its normalized expansion. It follows that every exterior support to \(A_z\) has expansion at most \[ k'+Cz, \tag{199}\] where \(C\) is independent of that support’s curvature.

The exact geometric feature of this application is worth recording. For a nearest pair \(x\in A\), \(x'\in\partial A_z\), let \(P:T_xM\to T_{x'}M\) be parallel transport along the short minimizing segment. The Jacobi construction in the common proof gives \[ F_z(x)=x',\qquad \,\mathrm dF_z|_x=P, \qquad |\nabla\,\mathrm dF_z|_x\le Cz. \tag{200}\] If \(\psi\) defines the support at \(x'\), then \(\psi\circ F_z\) supports \(A\) at \(x\), and \[ \begin{split} \mathop{\mathrm{Hess}}(\psi\circ F_z)_x(X,Y) ={}&\mathop{\mathrm{Hess}}\psi_{x'}(PX,PY)\\ &+\,\mathrm d\psi_{x'}\bigl((\nabla\,\mathrm dF_z)_x(X,Y)\bigr). \end{split} \tag{201}\] The exact tangent isometry preserves the normalized Hessian trace; only the \(O(z)\) second-map derivative and the \(C^1\) change of \(K\) enter the expansion error. Thus no term multiplying \(\mathop{\mathrm{Hess}}\psi\) has been discarded. This local transport step uses compact ambient bounds, while the following enclosure argument still uses the auxiliary total regions specific to the maximal end.

Fix \(k'<c'<0\), and take \(\beta>0\) below the preceding injectivity and collar scales, so small that \(k'+C\beta<c'\). Smooth approximation of the distance function, followed by a regular-value choice, gives a smooth compact region \(U\) with \[A_{\beta/4}\subset\operatorname{int}U, \qquad U\subset A_{3\beta/4}.\] For example, approximate the distance uniformly within \(\beta/32\), and choose a regular value between \(3\beta/8\) and \(5\beta/8\). Choose a smooth nonnegative function \(\zeta\) equal to one near \(\partial U\) with \(\mathop{\mathrm{supp}}\zeta\subset\{\mathop{\mathrm{dist}}(\cdot,A)<\beta\}\). For sufficiently large finite \(\Lambda\), replacing the auxiliary threshold tensor by \[K_{c'}^{\rm aux}-\Lambda\zeta g^{\rm aux}\] makes \(\partial U\) strictly trapped: the added term decreases its expansion by \(3\Lambda\). Apply the total-region theorem to these smoothly modified complete data. Let \(\widetilde{\mathcal D}\) be the resulting closed filled total region and \(\widetilde B\) its smooth frontier. It contains \(U\). All cutoffs were placed outside the fixed enlargement above; outside the support of \(\zeta\), large spheres retain positive expansion. The maximum-radius argument confines \(\widetilde B\) to the uncut exterior. There it satisfies \[ \Theta_K(\widetilde B)=c'+3\Lambda\zeta\ge c'. \tag{202}\]

Let \(z=\mathop{\mathrm{dist}}(A,\widetilde B)\), the global minimum distance. Because \(\widetilde{\mathcal D}\) contains \(A_{\beta/4}\), one has \(z\ge\beta/4>0\). If \(z<\beta\), every path of length less than \(z\) starting in \(A\) remains on the inner side of \(\widetilde B\): its first crossing would contradict the definition of \(z\). Taking closures proves \(A_z\subset\widetilde{\mathcal D}\). At a pair attaining the distance, \(\widetilde B\) is therefore an exterior support to \(A_z\). Equations (199) and (202) imply \[c'\le\Theta_K(\widetilde B)\le k'+Cz \le k'+C\beta<c',\] a contradiction. Thus \(z\ge\beta\). Since this is the global minimum, \(\zeta\) vanishes at every point of \(\widetilde B\), including when the minimum equals \(\beta\). It follows that \(\Theta_K(\widetilde B)=c'\) everywhere. Its filled inner region is an admissible threshold-\(c'\) region for the original auxiliary data, so \(A\subset\widetilde{\mathcal D}\subset\mathcal D_{c'}\), as required. ◻

Proof. This is Proposition [four:sys:scalar-proposition], with its axis \(e=e_f\) and the same graph variables and density normalization. Its arbitrary trace \(\tau\) is zero here.

We record the differential relations needed in the boundary and drift calculations. Differentiating the implicit definition of \(Z\) gives \[ 3\,dZ=(3+pv)dt+H^f(w,\cdot). \tag{215}\] If \(k=-H^f-p(dt\otimes w^\flat+w^\flat\otimes dt)\) and \(Q=K-k\), then \[\begin{align*} V/u&=P_Qw+3A_d\mathop{\mathrm{grad}}t+Fw, \tag{216}\\ u^{-1}\mathop{\mathrm{div}}(uw) &=F+(1-\chi)j-\mathop{\mathrm{tr}}_gK+2(p+1)w(t). \tag{217}\end{align*}\] These are the identities in the proof of the cited proposition. They follow by inserting \(\mathop{\mathrm{tr}}T=F-\chi j\) and \(H^f(w,\cdot)=v\,d\log\sigma\) into its invariant formula; their smooth tensor expressions also hold at \(df=0\). The quadratic expression there is precisely the displayed \(\mathcal T\). ◻

Proof. Lemma [four:sys:coercivity-lemma] applies with the same dictionary. For clarity, its exact square completion is \[\begin{align*} \mathcal T={}&\tfrac12|T^0|^2+|M+(p+1)ay|^2 +\bigl(3(p+1)-v\bigr)|y|^2\\ &+3(p+1)d\left(x+\frac{\chi aj}{3d}\right)^2 +\tfrac23\left(F-\frac{\chi j}{4}\right)^2 +\frac{d(48-3d)}{8(3+pv)^2}j^2. \tag{220}\end{align*}\] The last coefficient is at least \(45d/[8(3+N)^2]\) and the coefficient of \(|y|^2\) is at least two. Undoing the shifted squares therefore costs only a polynomial in \(N\). When \(dt=0\), the normal terms instead reduce to \[\tfrac23(F-\chi j/4)^2+\chi(1-3\chi/8)j^2.\] The bounds \(5/8\le1-3\chi/8\le1\) and \(\chi^2\le\chi\) give the absolute equivalence claimed in (219). ◻

Proof. Apply Lemma [four:sys:boundary-lemma] with the chosen normal \(\nu\) and trace relation (211). Its formulas use the signed component \(w_\nu\) and extend smoothly to \(df=0\), so they also apply when the normal graph slope changes sign. ◻

Proof. At an interior extremum of \(h\), we have \(w=0\), \(A_\chi=\mathop{\mathrm{Id}}\), \(D_0=D_1=0\), and \(\mathop{\mathrm{tr}}K=0\). The trace equation and its positive Hessian coefficient show that a minimum has \(h\ge0\) and that a maximum has \(h\le\|C\|_\infty\). The boundary heights lie in \([b_*,0]\). Consequently \[ b_*\le h\le\|C\|_\infty \tag{235}\] in stages one through three; stage four has \(f=0\). All remaining terms in the trace prescriptions are uniformly bounded for \(|w|\le1\), \(0\le\chi\le1\), so \(F\) has a fixed bound as well.

Outside the compact supports the trace equation takes the form \[\beta\mathop{\mathrm{tr}}_{A_\chi}\mathop{\mathrm{Hess}}h+s_K\mathop{\mathrm{tr}}_{A_\chi}K=h, \qquad \beta=\frac{l}{\eta_N\sqrt D},\qquad 0\le s_K\le1.\] Here \(s_K=1\) in the first two stages and \(s_K=1-\zeta\) in the third. At a nonzero test gradient \(\beta=a/|\mathop{\mathrm{grad}}h|\). Test with \(\psi=C_2(r^{-\gamma}-R^{-\gamma})\). Since the test axis is radial, the end metric estimates give, uniformly for \(0<\chi\le1\), \[\frac{\mathop{\mathrm{tr}}_{A_\chi}\mathop{\mathrm{Hess}}\psi}{|\mathop{\mathrm{grad}}\psi|} =\frac{-3+(\gamma+1)\chi}{r}+O(r^{-1-q}) \le-\frac{c_1}{r},\qquad c_1=(2-\gamma)/2>0,\] after fixing \(r_0\) sufficiently large. In addition, maximality gives \[ |\mathop{\mathrm{tr}}_{A_\chi}K| \le K_0r^{-1-q}\min(1,(1+N/3)a^2). \tag{236}\] Indeed \(\mathop{\mathrm{tr}}_{A_\chi}K=(\chi-1)K(e_f,e_f)\) and \(1-\chi=(3+p)v/(3+pv)\le\min(1,(1+N/3)v)\). These estimates hold for every value of \(t\).

Put \(c_\gamma=1-2^{-\gamma}\) and \(H_0=\max(|b_*|,\|C\|_\infty)\). Choose the fixed coefficient \(C_2\) so that \[C_2\ge\max\left\{1, \frac{2H_0r_0^\gamma}{c_\gamma}, \frac{2K_0r_0^{\gamma-1-q}}{c_\gamma}\right\}.\] For \(R\ge2r_0\), this dominates the heights at \(r=r_0\). On \(r\le R/2\), the tensor term in (236) is at most \(\psi/2\), since \(\gamma<1+q\). On \(r\ge R/2\), use \(\min(1,(1+N/3)a^2)\le\sqrt{1+N/3}\,a\) and require \[R\ge2\left(\frac{2K_0\sqrt{1+N/3}}{c_1}\right)^{1/q}.\] The tensor error is then at most \(c_1a/(2r)\).

If \(h-\psi\) had a positive maximum, it would occur in the interior of this annulus. The gradients there agree and the Hessian of \(h\) is no larger than the Hessian of \(\psi\). The coefficients use the actual \(Z\) and this common gradient, so they agree at the contact regardless of the two heights. On the first half of the annulus the equation gives \(h\le-c_1a/r+\psi/2<\psi\); on the second half it gives \(h\le-c_1a/(2r)<0\). Both are contradictions. At a negative minimum of \(h+\psi\) the Hessian inequality reverses, while its axis and \(\chi\) are unchanged on reversing the gradient; the same estimates give the lower comparison. This proves the height bound in (234).

Because \(h=0\) at \(S_R\), the two comparisons bound its normal derivative there by \(C_3R^{-1-\gamma}\), with \(C_3=2C_2\gamma\) after increasing \(r_0\) so that \(|dr|\le2\). Tangential derivatives vanish, giving the asserted bound for \(|\mathop{\mathrm{grad}}f|\). Finally, at \(S_R\) the strictly increasing function \(G(t)=t+\frac16\log(1+l(t)^2\sigma^2)\) has \(G(t)=Z=0\). Since \(l(-\epsilon)=\ell\), its root exceeds \(-\epsilon\) whenever \(\ell^2\sigma^2<e^{6\epsilon}-1\). It suffices to require \[ R^{2\gamma+2}> \frac{C_3^2e^{\epsilon N}}{e^{6\epsilon}-1}. \tag{237}\] In stage four the conclusion follows instead from \(f=0\) and \(Z=0\) on \(S_R\). Every restriction on \(R\) made here depends only on the fixed data, \(N,\epsilon\), and is compatible with \(R\to\infty\) at fixed \(N\). ◻

Proof. Lemma [four:sys:gauss-lemma], with \(e=e_f\), uses exactly (239). At a minimum \(\mathop{\mathrm{Hess}}t\ge0\), so its stationary Gauss identity gives (238).

The two algebraic identities used by its floor estimate specialize to \[\begin{align*} \mathcal T-\mathcal S(S_f) ={}&|T^0|^2+(1+d)|M|^2\\ &+\chi(1+d-2\chi/3)j^2+(\chi/3-d)Fj+F^2/3 \tag{241}\end{align*}\] and \[ \frac{d^2}{\chi}=\frac{d(3+pv)}3\le\frac{N+3}3. \tag{242}\] They are [four:sys:stationary-difference] and the first ratio in [four:sys:polynomial-ratios]. The common proof splits at \((1+p)v\) small, uses stationary coercivity, and replaces \(S_f\) by \(H^f=S_f-K\). It yields an absolute positive \(c_2\) and a polynomial \(\Pi_N\) in precisely (240), including the smooth extension to \(df=0\). ◻

Proof. Differentiating \(w=l\mathop{\mathrm{grad}}f/\sqrt D\) covariantly gives \[ \nabla_iw=A_d(H^f_{i\cdot})+pd\,w\,\partial_i t, \qquad d\log u=(p+pv+2)dt+H^f(w,\cdot). \tag{246}\] In the first formula, \(d\) is the scalar \(D^{-1}\). Relative to the axis, the \(A_d\) factor multiplies the axial output of \(H^f\) by \(d\). The transverse and mixed entries are controlled by (218); the possibly large axial entry occurs as \(d(j-K_{ee})\) and is also controlled because \(d\le\sqrt d\). Furthermore \(d|dt|\le|dt|_{A_d}\). This proves the estimate for \(\nabla w\) without an inverse power of \(l\) or \(d\). For the second formula use \(\chi\le d\): the weighted norm of its first term is at most \((2N+2)|dt|_{A_d}\), while its last term is bounded by \(a(|M-K_{e\perp}|+\sqrt\chi\,|j-K_{ee}|)\).

Write \(A_\chi=\mathop{\mathrm{Id}}-c(p,v)w\otimes w\), where \(c(p,v)=(3+p)/(3+pv)\). The relevant derivatives are \[\partial_pc=\frac{3d}{(3+pv)^2},\qquad \partial_vc=-\frac{p(3+p)}{(3+pv)^2},\qquad |\nabla v|\le2a|\nabla w|.\] The \(p'\) contribution retains its factor \(d\), and is bounded by a polynomial times \(d|dt|\). All other terms are bounded by a polynomial times \(|\nabla w|\). This proves (243).

In (244), differentiating \(u\) contributes at most \[|d\log u|_{A_\chi}|K(w,\cdot)|_{A_\chi}.\] The other derivatives contribute at most \[|K|(|\nabla A_\chi|+|\nabla w|)+a|\nabla K|.\] Apply (243) and then Young’s inequality to \(\Pi_N|K|\sqrt\mathcal T\). Squaring a polynomial remains polynomial, proving (244) with the asserted dependence.

For (245), the common height term gives exactly \(w(h)=\eta_N a\sigma\). Derivatives of \(C,D_0,D_1\) cost, on \(\mathcal K\), a fixed multiple of \(1+|\nabla w|\), hence an arbitrarily small fixed multiple of \(\mathcal T\) plus a polynomial compact error and \(\Pi_N|K|^2\). The only other term is \(w(\mathop{\mathrm{tr}}_{A_\chi}K)\) in stage three; it is bounded in absolute value by a fixed multiple of \(a|\nabla K|+a|K||\nabla A_\chi|\), handled in the same way. All stage parameters lie in \([0,1]\), so the estimates are uniform in them. This proves the last assertion. ◻

Proof. We first record all end losses that will occur. There is a fixed constant \(C_*\) with \[ \mathbf1_{\mathcal K}+|K|^2+vF^2+v|\mathop{\mathrm{Ric}}_g|+a|\nabla K| \le C_*(\rho_0+v\rho_1). \tag{247}\] On a fixed compact set this follows from positivity of \(\rho_0\) and Lemma 47. Outside that set, \(|F|\le |h|+|K|\) in all three trace stages, and the same lemma controls \(h^2\) by a fixed multiple of \(r^{-2\gamma}\). The metric and tensor decay give \(|\mathop{\mathrm{Ric}}_g|=O(r^{-2-q})\), \(|K|^2=O(r^{-2-2q})\) and \(|\nabla K|=O(r^{-2-q})\). Since \(2\gamma<2+q\) and \(2+2q>4+\delta\), the curvature and squared tensor terms have the stated bounds. For the derivative term the precise estimate is \[ar^{-2-q} \le\tfrac12a^2r^{-2\gamma} +\tfrac12r^{-4-2q+2\gamma} \le\tfrac12v\rho_1+\tfrac12\rho_0,\] where \(2(q-\gamma)>\delta\) was used. This proves (247). Also \(\rho\le C_\rho\rho_0\) for a fixed \(C_\rho\), by its prescribed tail and compact positivity of \(\rho_0\).

Suppose first that the minimum occurs in the interior in one of the first three stages. Then \(dt=0\) and \(\mathop{\mathrm{Hess}}t\ge0\) there. Proposition 44 and (238) imply, using \(\mathop{\mathrm{tr}}K=0\), \[u^{-1}\mathop{\mathrm{div}}V \ge\mu-\tfrac12R_g+J(w)+v\mathop{\mathrm{Ric}}_g(e_f,e_f) +\mathcal T-\mathcal S(H^f)+w(F).\] Here \(\mu-R_g/2=-|K|^2/2\) and \(|J(w)|\le C_*a|\nabla K|\). Use (240), (245), and (247); then pass from \(V\) to \(V_\lambda=V-(1-\lambda)uA_\chi K(w,\cdot)\) by (244). Choose the two fixed Young errors smaller than \(c_2/4\). This yields an absolute \(c_3>0\) and a polynomial bound \(\Pi_N^{\mathrm f}\) such that at the proposed minimum \[ u^{-1}\mathop{\mathrm{div}}V_\lambda \ge c_3\mathcal T+\eta_N a\sigma -\Pi_N^{\mathrm f}(\rho_0+v\rho_1). \tag{248}\] In particular \(c_3\) is independent of \(N\).

Fix \(\delta_0<\min(c_3/2,1/4)\). At the floor \(m_0=1\), and the prescribed right side in the first three stages is \(\delta_0\mathcal T+\rho+\delta_0\eta_N a\sigma-C_N(\rho_0+v\rho_1)\). The lower bound in (248) exceeds it by at least \[(c_3-\delta_0)\mathcal T+(1-\delta_0)\eta_N a\sigma +(C_N-\Pi_N^{\mathrm f})(\rho_0+v\rho_1)-\rho.\] Choose \(C_N\ge\Pi_N^{\mathrm f}+C_\rho+1\). The last display is strictly positive because \(\rho_0>0\), contradicting the equation. This choice is polynomial and independent of \(R\) and of the solution.

In stage four use (233). At an interior floor minimum the left side is \(u^{-1}\mathop{\mathrm{div}}(3u\mathop{\mathrm{grad}}t)=3\Delta t\ge0\), while the right side is \[\eta\bigl(\delta_0|K|^2/2+\rho-C_N\rho_0\bigr).\] Increasing \(C_N\) by a fixed constant makes the bracket strictly negative everywhere, since \(|K|^2\le C_*\rho_0\). Thus an interior floor minimum is impossible whenever \(\eta>0\).

An inner boundary minimum in the first three stages is also impossible. At \(t=-\epsilon\) we have \(j_0=0\). Since \(V_\lambda=V-(1-\lambda)uA_\chi K(w,\cdot)\), the drift correction in (229) cancels, leaving \(V_\nu/u=\mathcal B\). Lemma 46 therefore gives \[3\partial_\nu t=-H-N_B<0.\] This contradicts the nonnegative derivative into \(\Omega_R\) required at a boundary minimum. In stage four, where \(f=0\), the corresponding formula is \(3\partial_\nu t=\eta(-H-N_B)<0\) for \(\eta>0\). The outer boundary is excluded by Lemma 47.

It remains to consider \(\eta=0\). The last-stage equation and boundary conditions then reduce to \[\mathop{\mathrm{div}}(3L_0(Z)\mathop{\mathrm{grad}}Z)=0,\qquad \partial_\nu Z|_B=0,\qquad Z|_{S_R}=0.\] Multiplying by \(Z\) and integrating gives \(\int_{\Omega_R}3L_0(Z)|\mathop{\mathrm{grad}}Z|^2\,\mathrm dV_g=0\). Thus \(Z=t=0\), which is strictly above the floor. This treats every stage and every possible location of a floor minimum. ◻

Proof. We first identify the geometric inequality satisfied by a relaxed limit. Take any sequence of admissible solutions with \(N_i\to\infty\) and allowed \(R_i\to\infty\), and write \[\underline h(x)=\lim_{j\to\infty} \inf\{h_i(y):i\ge j,\ \mathop{\mathrm{dist}}(x,y)<j^{-1}\}.\] On compact sets this is a finite lower semicontinuous function, by Lemma 47. Let a smooth function \(\phi\) touch \(\underline h\) from below at an interior point \(x\), with \(\mathop{\mathrm{grad}}\phi(x)\ne0\). Subtracting a small multiple of the fourth power of the distance makes the contact strict without changing its two-jet. Minimization on a small closed ball then gives a subsequence of local lower contacts of \(\phi\) plus constants with \(h_i\), at points \(x_i\to x\), whose contact values converge to \(\underline h(x)\).

At these contacts, \(\eta_{N_i}\mathop{\mathrm{grad}}f_i=\mathop{\mathrm{grad}}\phi\), so \[l_i\sigma_i\ge\frac{\ell_i}{\eta_{N_i}}|\mathop{\mathrm{grad}}\phi(x_i)|\longrightarrow\infty, \qquad d_i,\chi_i\longrightarrow0,\quad a_i\longrightarrow1.\] In particular this conclusion uses the floor and the fixed test gradient, rather than an estimate for the solution gradients. Put \(\nu_\phi=\mathop{\mathrm{grad}}\phi/|\mathop{\mathrm{grad}}\phi|\). Positivity of \(A_\chi\) and the Hessian inequality at a lower contact give \[\mathop{\mathrm{tr}}_{A_{\chi_i}}K+ \frac{a_i}{|\mathop{\mathrm{grad}}\phi|} \mathop{\mathrm{tr}}_{A_{\chi_i}}\mathop{\mathrm{Hess}}\phi \le F_i\le h_i.\] The last inequality holds throughout stages 1 and 2 because \(C\le0\) and \(D_0\le0\). Passing to the limit proves \[ \frac{\mathop{\mathrm{tr}}_{\nu_\phi^\perp}\mathop{\mathrm{Hess}}\phi}{|\mathop{\mathrm{grad}}\phi|} +\mathop{\mathrm{tr}}_{\nu_\phi^\perp}K\le\underline h(x). \tag{250}\] The left side is the outward expansion of the regular level of \(\phi\). Neither convergence of the homotopy parameters nor continuity bounds for the solution coefficients were needed.

Fix \(c<k<k'<0\). Apply Lemma 21 on \(U=\operatorname{int}\Omega\) with \(u=\underline h\), \(Q=K\) and \(q_0=k'\). The relaxed limit is finite and lower semicontinuous. At a nonzero-gradient lower test whose contact value is less than \(k'\), (250) bounds its expansion by that value, hence by \(k'\). The lemma therefore shows that every smooth exterior support to the closed sublevel \(\{\underline h\le k\}\) at an interior frontier point has expansion at most \(k'\). In particular, this conclusion also applies when the sublevel has a plateau at height \(k\).

Lemma 47 makes this sublevel bounded: for a large fixed \(r\), its lower height bound exceeds the fixed negative number \(k\). Adjoin \(\mathcal D_c\) to obtain a compact closed set. At a frontier point on \(B\), each exterior support also supports \(\mathcal D_c\); tangency comparison with its smooth boundary gives expansion at most \(c\). Lemma 43 therefore places the union in \(\mathcal D_{c'}\) for every \(k'<c'<0\).

Hausdorff right continuity at \(c\) and the positive distance of \(Q_0\) from \(\mathcal D_c\) allow us to choose \(c<k<k'<c'<0\) so that \(\mathcal D_{c'}\cap Q_0=\varnothing\). If (249) failed, choose \(N_i\to\infty\), allowed \(R_i\ge i\), and \(x_i\in Q_0\) with \(h_i(x_i)\le c+i^{-1}\). A subsequence has \(x_i\to x\in Q_0\), and then \(\underline h(x)\le c<k\), contradicting that containment. This also proves uniformity over the two stages. Enlarging the allowed radius function never invalidates the assertion. ◻

Proof. Use a fixed short collar \(0\le s\le s_0\) from Lemma 42, with \(C=-\kappa s\), \(\nu_s=\mathop{\mathrm{grad}}s/|\,\mathrm ds|\), and \(0<c_s\le|\,\mathrm ds|\le C_s\). All its geometric derivatives are bounded, and \(P_s+H_s\ge c\), where \(P_s=\mathop{\mathrm{tr}}_{\{s=\mathrm{constant}\}}K\). The finitely many outer collar faces satisfy Lemma 51.

For a positive-slope test \(h_0=c+\psi(s)\), the trace operator, with \(t\) computed from the solution’s \(Z\) and the common contact gradient, is exactly \[ P_s+aH_s+\chi K(\nu_s,\nu_s) +a\chi\left( \frac{\mathop{\mathrm{Hess}}s(\nu_s,\nu_s)}{|\,\mathrm ds|} +|\,\mathrm ds|\frac{\psi''}{\psi'}\right). \tag{252}\] For a test with negative slope, replace \(a\) by \(-a\) in this formula. If \(|\psi'|\) lies between fixed positive constants, the floor gives \[ d\le C(\eta_N/\ell)^2=Ce^{-\epsilon N},\qquad 1-a\le d,\qquad \frac{N\chi}{d}=\frac{3N}{3+pv}\ge\frac{3N}{3+N}. \tag{253}\] Thus \(a\ge1/2\) and \(N\chi\ge c_4d\) for large \(N\).

Choose a fixed smooth \(\vartheta_1:[0,\infty)\to[0,1]\) equal to one on \([0,1]\) and zero on \([2,\infty)\), and put \(b_N(s)=\Lambda_* N\vartheta_1(Ns)\), where \(\Lambda_*>0\) is a fixed collar constant, independent of \(N\). The constant will be chosen large; its crucial property is \[0\le\int_0^{s_0}b_N(s)\,\mathrm ds\le2\Lambda_*.\] For the lower barrier in stages 1 and 2, prescribe \(\psi'(s)=q_-\exp(\int_0^s b_N)\) and \(\psi(0)=0\). For the chosen value of \(\Lambda_*\), take \(q_->0\) so small that \(\psi'\le\kappa/2\) and \(\psi(s_0)<\eta_2\). Positive normal direction makes \(D_0=0\). Subtracting the prescribed trace right side from (252) gives a residual at least \[ \kappa s/2-Cd+a\chi|\,\mathrm ds|b_N(s). \tag{254}\] The constant \(C\) in this residual depends only on collar geometry. We choose \(\Lambda_*\) so large that the last term dominates \(2Cd\) on \([0,1/N]\). On \([1/N,s_0]\), the first term dominates \(Cd\) for large \(N\), because \(Ne^{-\epsilon N}\to0\), and the last term has a favorable sign. This is a strict lower barrier. Its outer value is below \(h\) by separation and its inner value agrees with \(h\).

For the upper barrier take \(\psi'(s)=q_+\exp(-\int_0^s b_N)\). Choose its fixed minimum slope larger than twice the bounded linear variation of \(P_s+H_s-c+\kappa s\), and choose \(q_+\) still larger if needed to dominate the height bound at \(s_0\). The residual is now at most \(-c_7s+Cd-a\chi|\,\mathrm ds|b_N\), for fixed \(c_7>0\). The same two intervals prove strict negativity. The bounded integral of \(b_N\) leaves all lower and upper test slopes between fixed positive constants, independently of \(N\).

For completeness, at an interior positive maximum of \(h_0-h\), the test Hessian is at most the solution Hessian, whereas its height is larger. The positive principal matrix and the strictly increasing dependence of \(F\) on \(h\) contradict a positive test residual. This proves lower comparison; reversing the difference proves upper comparison. In both comparisons \(Z\) and the gradient are the same at contact, so all coefficients other than the height agree. Taking inward derivatives at \(s=0\) gives the first assertion of (251).

In stage 3 the face height is \(b_\zeta=(1-\zeta)c\). Use \(b_\zeta\pm\psi(s)\) with fixed large positive slopes and \(\psi''/\psi'=-b_N\). At \(s=0\) the two unit-direction inequalities in (232) say that the positive-slope residual is nonpositive and the negative-slope residual is nonnegative. Replacing \(\pm \nu_s\) by \(\pm a\nu_s\) costs \(O(d)\), including the term \(\chi K(\nu_s,\nu_s)\); moving to distance \(s\) costs \(O(s)\), uniformly in \(\zeta\). The change of height then supplies a strict linear margin, with the desired sign, when the minimum test slope exceeds the fixed spatial error constant. The term involving \(b_N\) has the correct sign for both tests and dominates the \(O(d)\) error on \([0,1/N]\). On the remainder the linear margin dominates it. The height bound handles \(s=s_0\). These two comparisons bound the slope in absolute value in stage 3. The terminal trace equation has \(f=0\), as established in Section 12. ◻

Proof. Set \(W=l\mathop{\mathrm{grad}}_{\bar g}f=\sqrt d\,w\). For each covector \(\xi\), \[|\xi(W)|\le|\xi|_{A_d} \le\sqrt{(3+N)/3}\,|\xi|_{A_\chi},\qquad uW=L_0w.\] The second inequality follows from \(d/\chi=(3+pv)/3\). Lemma 52 gives \(w\cdot\nu=a\ge1/2\) on \(B\). Direct differentiation yields \[\begin{align*} \mathop{\mathrm{div}}w&=\mathop{\mathrm{tr}}_{A_d}H^f+dp\,w(t),\\ u^{-1}\mathop{\mathrm{div}}(uW) &=\sqrt d\bigl(\mathop{\mathrm{tr}}_{A_d}H^f+(dp+p+2)w(t)\bigr),\\ D^{-1/2}\mathop{\mathrm{div}}(\sqrt D W) &=\sqrt d\bigl(\mathop{\mathrm{tr}}_{A_d}H^f+dp\,w(t)\bigr). \end{align*}\] These last two expressions have absolute value at most \(\Pi_N(1+\sqrt{\mathcal T})\) on the collars. Indeed the axial tensor term is \(d^{3/2}H^f_{ee}\), controlled by the \(d j^2\) term of Lemma 45, and \(\sqrt d\,|w(t)|\le|\,\mathrm dt|_{A_d}\); all background \(K\) terms are bounded there. No inverse power of \(l\) has occurred.

Take a fixed collar cutoff \(\eta\), equal to one near \(B\) and zero near the outer collar faces. Apply the divergence theorem to \(\eta\varphi uW\). The domain’s outward normal on \(B\) is \(-\nu\), so the boundary integral is \(-\int_B L_0a\varphi\). Taking absolute values of the three interior terms, and using \(a\ge1/2\), proves (255). The derivative of \(\eta\) costs only a fixed multiple of \(\varphi\). Using \(\sqrt D W=w\) instead proves (256). The same computation proves the asserted localization.

On \(B\), positive slope gives \(w_\nu=a\), \(D_0=0\), and \(F=c=P_B+H\). Thus the expression inside the boundary multiplier of the homotopy is \[(a-1)H-N_Bd-(1-\lambda)\chi aK(\nu,\nu).\] The multiplier lies in \([0,1]\), and \(1-a\le d\), \(\chi\le d\). This proves the first inequality in (257). The lower collar slope gives \(\sqrt d\le(l|\partial_\nu f|)^{-1}\le C\eta_N/\ell\), proving the rest. Finally apply (255) with \(\varphi=1\). On the fixed collars, \(\rho\) has a positive minimum, so \(1+\sqrt{\mathcal T}\le C(\rho+\delta_0\mathcal T)\) with fixed \(C\). This proves (258). ◻

Proof. We first finish stage 1. Apply (263) to \(\omega=\psi e^{kZ}\) and then use (264). If \(\psi=1\) on a smaller patch and \(|\,\mathrm d\psi|\le C/\mathrm{gap}\), the resulting norm step is \[\|e^Z\|_{L^{4k}(\text{smaller graph})} \le\bigl[C(N)k^2(1+\mathrm{gap}^{-2})\bigr]^{1/(2k)} \|e^Z\|_{L^{2k}(\text{larger graph})}.\] Choose \(k_j=2^j k_*(N)\) and geometrically decreasing gaps between fixed nested base patches. The logarithms of the factors sum because \(\sum_j (1+j)/2^j<\infty\). Writing \(S(r)=\sup_{Q_r}e^Z\) for a nested family of compact patches gives \[S(r')\le C(N)(r''-r')^{-A(N)} \|e^Z\|_{L^{2k_*}(\Gamma\cap\pi^{-1}(Q_{r''}))}.\] Interpolation with (262) yields \[\|e^Z\|_{2k_*} \le S(r'')^{1-1/k_*}\|e^Z\|_2^{1/k_*}.\] If \(k_*=1\) this already proves the bound. Otherwise Young’s inequality gives, for every \(\alpha>0\), \[S(r')\le\alpha S(r'') +C(N,\alpha)(r''-r')^{-A'(N)}.\] Iterate over radii increasing geometrically to a fixed larger radius and take \(\alpha<2^{-A'(N)-1}\). The sum of the remainder terms is finite. The final term \(\alpha^j S(r_j)\) tends to zero for each individual smooth solution, since all \(Q_{r_j}\) lie in a fixed compact patch. The resulting bound is independent of that individual supremum. Finitely many such patches prove an upper bound for \(Z\) on any fixed compact base set, including \(B\).

To extend the bound along the end, use (244), with a fixed small coefficient: \[u^{-1}|\mathop{\mathrm{div}}(uA_\chi K(w,\cdot))| \le\frac{\delta_0}{4}\mathcal T +\Pi_N(|K|^2+a|\mathop{\mathrm{grad}}K|).\] Since \(a\sigma= l\sigma^2/\sqrt D\ge e^{-1}a^2\), Young’s inequality bounds the second term by \[\frac{\delta_0}{4}\eta_N a\sigma +C(N)(|K|^2+|\mathop{\mathrm{grad}}K|^2).\] The last expression, apart from its favorable term, is \(o(\rho)\) on the end. Indeed \(2+2q>4+\delta\) and \(|\mathop{\mathrm{grad}}K|^2=O(r^{-4-2q})\). Beyond a sufficiently large fixed-\(N\) sphere, the high-level source (259) therefore implies \[\mathop{\mathrm{div}}(3uA_\chi\mathop{\mathrm{grad}}Z)>0 \quad\hbox{where }Z>Z_0(N).\] At an interior maximum its left side is \(3uA_\chi:\mathop{\mathrm{Hess}}Z\le0\), a contradiction. The compact bound controls the inner sphere and \(Z=0\) on \(S_R\). This proves the global upper bound in stage 1.

In stages 2 and 3, \(\lambda=0\). The same high-level source is strictly positive, so it excludes interior high maxima directly. The boundary slope bound implies \[Z=t+\tfrac16\log(1+l^2|\partial_\nu f|^2) \le t+\tfrac16\log(1+e^2 C_6^2/\eta_N^2) \quad\hbox{on }B.\] Thus sufficiently high boundary \(Z\) forces \(t\ge1\). At such points \(j_0(t)=1\), and the boundary prescription gives zero conormal flux. The boundary point principle excludes a nonconstant high maximum there: in a neighborhood of the maximum the individual smooth operator is elliptic and \(\mathop{\mathrm{div}}(3uA_\chi\mathop{\mathrm{grad}}Z)>0\), so its outward conormal derivative at a boundary maximum is strictly positive, contrary to the zero datum. This uses ellipticity only for each individual solution. The outer Dirichlet value is zero, so the upper bound follows uniformly through these stages.

In stage 4, \(f=0\) and \(t=Z\). For positive scaling \(\eta\), sufficiently high levels lie outside the penalty band and have source \(\eta(\delta_0\mathcal T+\rho)>0\); the same maximum and boundary point argument applies. At zero scaling, multiplying the homogeneous equation by \(Z\) gives \(\int 3L_0|\,\mathrm dZ|^2=0\), and the outer value yields \(Z=0\). These bounds are uniform even as \(\eta\downarrow0\).

Finally the height bound is Lemma 47, and \(t\le Z\) follows from \(D\ge1\). The implicit equation gives \[D=e^{6(Z-t)}\le e^{6(C(N)+\epsilon)},\qquad |\mathop{\mathrm{grad}}f|^2=(D-1)/l^2\le\ell^{-2}e^{6(C(N)+\epsilon)}.\] This proves (265). ◻

Proof. Write the trace prescription at parameter \(s\) as \[F_s=\eta_N f+F_s^0(x,Z,\mathop{\mathrm{grad}}f),\qquad Q=\frac{\sqrt D}{l},\qquad G_s=Q\bigl(F_s-\mathop{\mathrm{tr}}_{A_\chi}K\bigr).\] Here \(F_s^0\) is smooth in its displayed finite-dimensional arguments. For example, in stage three it is \((1-\zeta)(C+D_0)+\zeta(\mathop{\mathrm{tr}}_{A_\chi}K+D_1)\). The functions \(D_0,D_1\) are evaluated on \(w\), whose length is less than one. Thus \(F_s^0-\mathop{\mathrm{tr}}_{A_\chi}K\) is bounded on the finite truncation, uniformly in the gradient and in \(s\). The normalized scalar equation is \[ A_\chi:\mathop{\mathrm{Hess}}f=G_s(x,Z,f,\mathop{\mathrm{grad}}f). \tag{277}\] Its matrix is independent of the value of \(f\), and \(\partial_fG_s=\eta_N Q>0\).

For the scalar method of continuity introduce a further parameter \(\theta\in[0,1]\), retaining the assigned Dirichlet values and using \[\begin{align*} A^{(\theta)}&=(1-\theta)\mathop{\mathrm{Id}}+\theta A_\chi =\mathop{\mathrm{Id}}-(1-\chi_\theta)e_f\otimes e_f, &\chi_\theta&=1-\theta+\theta\chi,\tag{278}\\ G_s^{(\theta)}&=(1-\theta)\eta_N f+\theta G_s, &A^{(\theta)}:\mathop{\mathrm{Hess}}f&=G_s^{(\theta)}. \tag{279}\end{align*}\] At zero gradient these expressions are interpreted by their smooth tensor formulas. In particular, \[ \partial_fG_s^{(\theta)} =\eta_N\bigl(1-\theta+\theta Q\bigr)>0. \tag{280}\] Every trace stage has coefficient one on \(h=\eta_N f\), so this strict sign holds throughout both parameters.

Proof. All estimates from Sections 12–14 were conditional estimates for arbitrary smooth homotopy solutions. Specifically, Lemma 47 and Proposition 50 precede the bounded-range and derivative estimates; Lemma 51 and Lemma 52 apply to solutions satisfying the floor; Proposition 54 then bounds their variables; and Proposition 56 bounds their derivatives. None presupposes a fixed point of the map just defined. On this finite truncation these results give, uniformly in \(s\), \[\|Z\|_{C^{1,\alpha}}\le M_* \quad\text{for every smooth fixed point with }\min t\ge-\epsilon.\] The arbitrary exponent \(\alpha<1\) is permitted because the classical estimates give a \(C^2\) bound, and higher bounds as needed. Choose \(M_0>M_*+1\).

Let \[ \begin{split} m(s,Z)&=\min_{\overline\Omega_R}t\bigl(x,Z,\mathop{\mathrm{grad}}f_s[Z]\bigr),\\ \mathcal W&=\{(s,Z)\in[0,4]\times X: m(s,Z)>-\epsilon, \|Z\|_{C^{1,\alpha}}<M_0\},\\ \mathcal W_s&=\{Z:(s,Z)\in\mathcal W\}. \end{split} \tag{297}\] The minimum is continuous by Proposition 57, so \(\mathcal W\) is relatively open and every section is a bounded open subset of \(X\). Consider \[\mathcal K= \{(s,Z)\in\overline{\mathcal W}:Z=T_sZ\}.\] This is compact: any sequence has a convergent parameter subsequence, and joint compactness of \(T_s\) on the bounded input ball makes its fixed points converge in \(X\); continuity preserves the fixed-point equation and membership in the closed set. Every point of \(\mathcal K\) is a smooth solution with \(m(s,Z)\ge-\epsilon\). Its norm is at most \(M_*<M_0\), and Proposition 50 excludes \(m(s,Z)=-\epsilon\). Therefore \[ \mathcal K\subset\mathcal W. \tag{298}\]

Apply Lemma 31 with \(I=[0,4]\), the Banach space \(X\) above, the jointly compact homotopy \((s,Z)\mapsto T_sZ\), and \(\mathcal U=\mathcal W\). The uniform norm bound makes \(\mathcal W\) bounded; continuity of \(m\) makes it relatively open. The preceding compactness argument and (298) verify that its closed fixed-point set is compact and contained in \(\mathcal W\), so no fixed point lies on the relative boundary. The lemma gives constancy of \[\deg(\mathop{\mathrm{Id}}-T_s,\mathcal W_s,0)\qquad (0\le s\le4).\] It applies to these varying sections without a convexity assumption.

At \(s=4\) the scalar boundary values are zero and the final trace equation is \[\mathop{\mathrm{tr}}_{A_\chi}H^f=\eta_N f+D_1(x,w).\] At a positive or negative extremum \(w=0\) and \(D_1(x,0)=0\), so the strict height sign forces \(f=0\). This holds for every trial \(Z\), not just at fixed points. Thus \(t=Z\), \(\chi=1\) and \(u=L_0(Z)\). Both the drift and the prescribed source and boundary flux in (294) vanish. Testing its homogeneous equation by \(\widetilde Z\) gives \[\int_{\Omega_R}3L_0(Z)|\mathop{\mathrm{grad}}\widetilde Z|^2=0,\] so \(T_4Z=0\) for every input. At its sole fixed point \(Z=0\) one has \(t=0\), so \(0\in\mathcal W_4\) and \(\deg(\mathop{\mathrm{Id}}-T_4,\mathcal W_4,0)=1\) (Leray and Schauder 1934, II, §8, p. 55). The same degree at \(s=0\) yields a desired fixed point in \(\mathcal W_0\), proving the assertion on the finite truncation.

We finally make the parameter order explicit. Fix the strict background, \(\epsilon\), the collars and the end weights first. The pointwise floor argument chooses the small \(\delta_0\) and a polynomially large penalty coefficient \(C_N\) without derivative estimates. The separation statement is uniform along arbitrary expanding sequences of the first two homotopy stages; applied at the finitely many fixed collar ends, it supplies the eventual separation needed for the slopes. Increase \(N_0\) to meet this requirement and the finitely many conditions of the form \(\Pi_N\eta_N/\ell<\text{constant}\). Increase \(R_{\min}(N)\) to meet all the fixed-\(N\) outer-radius requirements, requiring also \(R_{\min}(N)\ge N\). These choices give the estimates for every \(N\ge N_0\) and every \(R\ge R_{\min}(N)\). All subsequent scalar, degree-radius and classical regularity constants may depend arbitrarily on the now fixed \(N\).

For that fixed \(N\), take \(R_j\to\infty\) and the desired fixed points just obtained. Proposition 56 gives uniform bounds of every finite order on each fixed compact subset, including patches meeting \(B\). Successive compact extractions on a countable exhaustion produce a subsequence converging in \(C^k\) on each such compact set for every finite \(k\). All equations, inner data and collar inequalities pass to the limit. The strict floor may become non-strict, which is the asserted \(t\ge-\epsilon\). The normalization at infinity is supplied by the uniform tail estimates in Proposition 59; it is not inferred from compact convergence alone. Only after this fixed-\(N\) exhaustion will \(N\) tend to infinity in the flux comparison. The latter uses the earlier polynomial estimates directly, and does not use the arbitrary constants of the present section. ◻

Proof. The local compactness and the inner boundary conditions follow from Propositions 56 and 58. To determine the value at infinity, apply Lemma 38 to the finite truncations before taking their local limit. We verify its hypotheses explicitly.

The strict background has \(1<q<2\), \(0<\delta<q-1\), \(g-\delta_{\mathrm{Eucl}}=O_2(r^{-q})\), \(K=O_1(r^{-1-q})\), \(\mathop{\mathrm{tr}}_gK=0\), and \(R_g=O(r^{-4-\delta})\) by Proposition 41. Outside the compact support of \(C\), the trace right side is \(\eta_N f\). The divergence equation is exactly (169), with \(\rho_0=r^{-4-\delta}\), \(\rho_1=r^{-2\gamma}\) and the smooth penalty switch defined in (224). Their differentiated end bounds follow from the fixed smooth radial choices. Propositions 54 and 56 supply the common bounded range and all fixed-\(N\) classical estimates, on end unit patches and on the zero-Dirichlet outer faces, uniformly in the truncation radius. The leading matrix is uniformly elliptic on that range.

The lemma therefore gives exponential decay of \(f\) and its fixed derivatives, a truncation-uniform bound \(|Z|+|t|\le C(N)r^{-2}\), and \(Z,t=O_2(r^{-2})\) for every local limit. In particular normalization at infinity follows from a comparison on the actual truncations, not from local convergence. The equation used for that comparison is \[\begin{align*} 3\Delta_g Z &+3\bigl[p(Z)+2-\delta_0(p(Z)+1)\bigr]|dZ|^2\\ &=\tfrac{\delta_0}{2}|K|^2+\rho -m_0(Z)C_N\rho_0+\mathcal E_N, \tag{301}\end{align*}\] where the graph error \(\mathcal E_N\) has exponential unit-patch bounds. The \(|K|^2/2\) term survives because the tensor is retained; its decay is integrable since \(2+2q>4+\delta\). All constants in this application are allowed to depend on the fixed \(N\). They will not be used to estimate the later limit in \(N\).

The signs can now be recorded directly from the desired system. Since \(F=\eta_N f+C\) and \(w(f)=a\sigma\), the scalar identity becomes \[\begin{align*} \tfrac12e^{2t}R_{\widehat g} ={}&[\mu+J(w)+w(C)-\rho] +(1-\delta_0)\mathcal T\\ &+(1-\delta_0)\eta_N a\sigma+m_0(t)\mathcal P. \tag{302}\end{align*}\] The bracket is nonnegative by \(|w|\le1\) and \(\mu-|J|\ge|dC|+\rho\); all remaining terms are nonnegative. The boundary identity and the prescribed conormal datum give \[ H_B+3\partial_\nu t=-N_B, \qquad \widehat H_B=-e^{-t}\sqrt d\,N_B<0. \tag{303}\] This uses the normal into \(\Omega\) on every component of \(B\).

As quadratic forms \(\widehat g\ge e^{-2\epsilon}g\). A \(\widehat g\)-Cauchy sequence is therefore \(g\)-Cauchy; completeness of the background exterior and local equivalence of the smooth metrics give completeness with the boundary included. Since \(q<2\), the conformal \(O_2(r^{-2})\) term and the exponential graph error retain the background decay \(\widehat g-\delta=O_2(r^{-q})\). Equation [four:maximal:end:zero-graph] gives \(\Delta_g t=O_N(r^{-4-\delta})\), since \(|dt|^2=O_N(r^{-6})\) and \(0<\delta<1\). Finally, \[R_{\widehat g} =e^{-2t}\bigl(R_{\bar g}-6\Delta_{\bar g}t -6|dt|_{\bar g}^2\bigr)\] and the exponential graph errors, together with \(R_g=O(r^{-4-\delta})\), give the scalar decay in (300). ◻

Proof. The end estimates imply \[\begin{align*} \mathcal T&=\tfrac12|K|^2+3(p(t)+1)|dt|^2+O_N(e^{-cr}),\\ u&=1+O_N(r^{-2}),\qquad V=3\mathop{\mathrm{grad}}_g t+O_N(r^{-5})+O_N(e^{-cr}). \end{align*}\] In particular \(u\Xi\) is absolutely integrable. Its nonexponential terms have decay \(r^{-2-2q}\), \(r^{-6}\), or \(r^{-4-\delta}\). The possibly slower weight \(\rho_1\) is multiplied by the exponentially small \(v\) on this fixed-\(N\) end. The divergence theorem gives \[\int_{S_r}V\cdot\nu_g\,\mathrm dA_g -\int_B V\cdot\nu\,\mathrm dA_g =\int_{\Omega_r}u\Xi\,\mathrm dV_g.\] Here \(\Omega_r\) denotes the part of \(\Omega\) inside \(S_r\). Absolute integrability proves existence of the limit and yields the orientation-sensitive formula \[ \mathfrak F=\int_\Omega u\Xi\,\mathrm dV_g +\int_B V\cdot\nu\,\mathrm dA_g. \tag{306}\]

We next compute the mass change in the same asymptotic chart. The linear conformal perturbation is \(2t\delta_{ij}\), and in dimension four its contribution to the ADM numerator is exactly \[\partial_j(2t\delta_{ij})-\partial_i(2t\delta_{jj}) =-6\partial_i t.\] The differentiated remainders are \(O_N(r^{-3-q})+O_N(r^{-5})+O_N(e^{-cr})\) and have vanishing sphere integrals. They include products of \(t\) with \(g-\delta\), quadratic conformal terms, and the graph perturbation. Replacing Euclidean surface flux for \(dt\) by \(g\)-surface flux changes it by \(O_N(r^{-q})\) after integration. Since the ADM denominator is \(6\omega_3\), we obtain \[E_{\widehat g}-E_g =-\frac1{\omega_3}\lim_{r\to\infty} \int_{S_r}\partial_{\nu_g}t\,\mathrm dA_g =-\frac{\mathfrak F}{3\omega_3}.\] This also proves existence of the ADM energy of \(\widehat g\) from the assumed existence of \(E_g\); no interchange of the \(r\) and \(N\) limits is involved.

To estimate (306), use the boundary flux bound (257) and the weighted trace estimate (255) with test function one. They give \[\begin{align*} \int_B|V\cdot\nu|\,\mathrm dA_g &\le C\frac{\eta_N}{\ell}\int_B L_0\,\mathrm dA_g\\ &\le\Pi_N\frac{\eta_N}{\ell} \int_{\mathrm{collars}}u(1+\sqrt{\mathcal T})\,\mathrm dV_g\\ &\le\Pi_N\frac{\eta_N}{\ell} \int_{\mathrm{collars}}u(\rho+\delta_0\mathcal T)\,\mathrm dV_g. \tag{307}\end{align*}\] For the last step, \(\rho\) has a positive minimum on these fixed compact collars, and \(\sqrt{\mathcal T}\le c\mathcal T+C_c\) with fixed \(c>0\). Only the polynomial constants in those two earlier estimates enter [four:maximal:end:boundary-cost]. Since \(\eta_N/\ell=e^{-\epsilon N/2}\), increasing \(N\) absorbs this entire cost into, for example, one half of the \(u(\rho+\delta_0\mathcal T)\) contribution to the bulk integral.

The negative bulk term is \(um_0(t)\mathcal P\). On its support, \[-\epsilon\le t\le-\epsilon+1/N, \qquad \ell\le l\le e\ell,\qquad c\ell\le L_0\le C\ell.\] These constants are uniform for \(0<\epsilon<1\) and sufficiently large \(N\). From the definitions, rather than from a derivative estimate, one obtains \[ u\le C(\ell+\ell^2\sigma),\qquad uv\le C\ell^2\sigma,\qquad ua\sigma=L_0l\sigma^2\ge c\ell^2\sigma^2. \tag{308}\] For the middle inequality use \(v\sqrt D=l^2\sigma^2/\sqrt{1+l^2\sigma^2}\le l\sigma\). Thus Young’s inequality, with \(C_N\) polynomial and \(0\le m_0\le1\), gives \[\begin{align*} um_0\mathcal P &\le \Pi_N\bigl[\ell\rho_0+ \ell^2\sigma(\rho_0+\rho_1)\bigr]\\ &\le\tfrac12\delta_0\eta_N ua\sigma +\Pi_N\left[\ell\rho_0+ \frac{\ell^2}{\eta_N}(\rho_0^2+\rho_1^2)\right]. \tag{309}\end{align*}\] Off the band the left side vanishes, so the final inequality holds everywhere. All weights in the loss are integrable: \[\rho_0=r^{-4-\delta}\in L^1,\qquad \rho_0^2\in L^1,\qquad \rho_1^2=r^{-4\gamma}\in L^1\] on the four-dimensional end, using \(\delta>0\) and \(\gamma>1\). Their smooth extensions are bounded on the compact interior. In particular no integrability of \(\rho_1\) itself is being used.

Insert [four:maximal:end:boundary-cost] and [four:maximal:end:penalty-loss] into (306), absorb the indicated nonnegative terms, and discard the remaining nonnegative bulk terms. This proves (305). Finally, \[\ell+\frac{\ell^2}{\eta_N} =e^{-\epsilon N}+e^{-\epsilon N/2},\] which tends to zero against every polynomial in \(N\). The constants used to prove (299) have not entered this estimate. ◻

Completion of Proposition 40. The threshold exterior and its all-cut comparison are supplied by Lemma 42. Propositions  58 and 59 construct the complete metric with the stated signs and decay. Its lower metric bound gives the asserted three-dimensional area comparison on each cut separately. Proposition 60 gives the energy estimate, since \(\ell=e^{-\epsilon N}\) and \(\eta_N=\ell^{3/2}\). Every stated output is therefore established before the Riemannian Penrose inequality is applied. ◻

Proof. For the classical codimension-one regularity background, see (Federer 1969, secs. 5.3–5.4) and (Federer 1970); the smooth-Riemannian formulation is (Simon 2014, chap. 7, §5, Theorem 5.8, p. 200). Apply Proposition [four:input:enclosure] to \((X,g_X)=(\Omega,\widehat g)\). The threshold construction gives a smooth connected orientable four-dimensional exterior with nonempty compact smooth boundary \(B\) and exactly one end. Proposition 59 gives completeness with \(B\) included and \(\widehat g-\delta=O_2(r^{-q})\) with \(q>1\). These are the full geometric hypotheses of the cited proposition; it imposes no scalar-curvature condition and does not require decay exponent two. Its perimeter counts the entire frontier, including contact with the obstacle.

The strict inequality \(\widehat H_B<0\) places the minimizing frontier \(\Sigma\) strictly outside \(B\) and makes it a nonempty compact smooth embedded minimal hypersurface. The same proposition gives its connected complete one-ended exterior and outer area-minimization with every component counted, including disconnected enclosing competitors.

Every such cut encloses the original obstacle. The cut comparison in Lemma 42 gives \(|\Sigma|_g\ge A_*\). Restriction of \(\widehat g\ge e^{-2\epsilon}g\) to each three-dimensional tangent plane gives \[|\Sigma|_{\widehat g}\ge e^{-3\epsilon}|\Sigma|_g \ge e^{-3\epsilon}A_*.\] This proves the asserted area comparison. The argument uses the general full-perimeter proposition directly, with its full \(q>1\) decay range. An enclosure result requiring \(O_2(r^{-2})\) decay would not suffice for this application. ◻

Proof of Theorem 39. First keep one set of strict approximation data fixed. For fixed \(\epsilon>0\) and each sufficiently large \(N\), take the exhaustion limit of Proposition 59. Apply (311) to the exterior of \(\Sigma\) from Lemma 61, with metric \(\widehat g\). That exterior is smooth and complete with boundary, has one end, and has nonnegative scalar curvature. Its decay exponents are \(p_0=q>1\) and \(s_0=4+\delta>4\) by (300). Its boundary is compact, minimal, and outer area-minimizing by the lemma. Its ADM energy exists by Proposition 60 and is unchanged by restriction past the compact region. Thus every hypothesis of the Riemannian theorem has been checked for this new exterior.

Equations (304), (305), and (310) give \[\begin{align*} E_g+\frac{\Pi_N}{3\omega_3} \left(\ell+\frac{\ell^2}{\eta_N}\right) &\ge E_{\widehat g} \ge\frac12\left(\frac{\widehat A}{\omega_3}\right)^{2/3}\\ &\ge\frac12e^{-2\epsilon} \left(\frac{A_*}{\omega_3}\right)^{2/3}. \end{align*}\] Let \(N\to\infty\) with these strict data and \(\epsilon\) fixed. The error vanishes by its polynomial bound. Letting \(\epsilon\downarrow0\) next yields \[E_g\ge\frac12\left(\frac{A_*}{\omega_3}\right)^{2/3}\] for the strict data. No convergence of the metrics \(\widehat g\) as \(N\to\infty\) has been assumed: only their individual inequalities and the uniform deficit are used.

Finally remove the strict approximation using Proposition 41. The approximation energies converge to the original \(E\), and their enclosing infima \(A_*\) converge to the original enclosing infimum. The given outer area-minimizing hypothesis, including the admissibility of \(S\) itself and all disconnected enclosing competitors, identifies this infimum with \(A=|S|_g\). Therefore \[\boxed{\displaystyle E\ge\frac12\left(\frac{A}{2\pi^2}\right)^{2/3}.}\] The prescribed \(P=0\) and \(E>0\) identify this energy with the ADM rest mass \(m=\sqrt{E^2-|P|^2}=E\). The original boundary was used with its future MOTS condition; no minimality or vanishing tangential trace of \(K\) on that boundary was imposed. ◻