Proof. Truncate the given end at a sufficiently large rotation sphere. Both boundary spheres carry ordinary effective rotations. Here is a direct verification of the assertion for an invariant sphere. Its Euler characteristic forces fixed points. At a fixed point the tangent representation is a planar rotation with integer weight. That weight has absolute value one: otherwise a nonidentity group element would fix the point and have identity differential both on the sphere and on its invariant inward normal, hence would be an isometry with identity first jet and therefore the identity on the connected ambient manifold. At a nonfixed point a finite stabilizer acts trivially on the orbit tangent. It acts trivially also on the transverse tangent line, by orientation, and on the inward normal. The same first-jet argument excludes such a stabilizer. The slice theorem now makes the quotient of the sphere an interval, with an ordinary rotation disk at either endpoint; the free circle bundle over its interior is a product. Consequently the action extends over an ordinary rotation ball.

Cap the two boundary spheres equivariantly, using invariant collars. The resulting closed three-manifold is simply connected. Its orbit space is a compact orientable surface with nonempty boundary, the boundary arising from fixed axes. The only other possible singular orbits are isolated exceptional fibers, represented by interior cone points of finite cyclic order. The orbifold fundamental group of this surface is a quotient of the fundamental group of the three-manifold. Indeed, away from axes and exceptional fibers, loops lift to paths; an endpoint displacement in a fiber can be closed within that fiber without changing the projected loop. Filling an axis tube introduces no nontrivial projected fiber relation. Filling an exceptional solid torus imposes the relation that the corresponding meridian raised to the isotropy order is trivial, as seen in its cyclic disk cover. Van Kampen therefore gives the asserted surjection onto the orbifold group.

An orientable surface with nonempty boundary and cyclic cone points has orbifold group equal to the free product of its ordinary free surface group and the cyclic groups at those points. Triviality of this group implies that the surface is a disk and that there are no cone points. Removing the two cap quotients gives the half-annulus description. Its regular interior is contractible, so its principal circle bundle is trivial.

For the last smoothness assertion, use equivariant normal disks near each axis and a signed transverse polar radius there. Changes of angular reference are smooth functions on the quotient, even in that signed radius. Their local phase lifts can be patched by a partition of unity; the only obstruction is the integral circle cocycle, which vanishes on this orbit rectangle. The same construction relative to a far-end strip preserves the given azimuth there. Thus the product gauge has the stated compatibility with the smooth axes. No isothermal, Weyl, or other metric coordinates have been assumed. ◻

Proof. The magnetic flux form is closed by \(\mathop{\mathrm{div}}_gB=0\). Since \(\Omega\) retracts onto a sphere, its only two-dimensional de Rham period is the magnetic flux, which is zero after rotation. The form is therefore exact. The decay can be imposed without sacrificing smoothness. On a rescaled annulus, radial homotopy reduces the primitive problem to a two-form on a fixed sphere; its zero integral is exactly the condition for a spherical primitive. These homotopy operators, followed by a fixed spherical elliptic inverse, have uniform bounds through the required derivatives. Scaling back a two-form of Cartesian order \(r^{-3}\) gives a primitive of order \(r^{-2}\), with its first two derivatives of the corresponding orders. On consecutive overlapping annuli, the difference of the primitives is a closed one-form and hence is the differential of a function. After subtracting a constant, its rescaled derivatives are bounded. Use cutoff multiples of these functions on the overlaps to glue the annular primitives. The annuli can be chosen with bounded overlap, so these corrections retain the bounds. The same argument glues the end primitive to any interior primitive. All ingredients are smooth on each finite annulus; this avoids any demand for uniform bounds on unspecified higher derivatives at infinity. Averaging over the circle makes \(\alpha\) invariant and preserves its equation and decay.

Set \(\chi=\alpha(\eta)\). Cartan’s formula gives \[\mathrm d\chi=-\iota_\eta(*_gB^\flat), \qquad (E\times B)\cdot\eta=-E\cdot\nabla\chi.\] The contraction of the momentum constraint with the Killing field, together with \(\mathop{\mathrm{div}}E=0\), now gives \[\mathop{\mathrm{div}}_g\big((K-\tau g)(\eta,\cdot)^\sharp+2\chi E\big)=0.\] The divergence theorem proves conservation of Equation (19). If \(\alpha\) is changed by a closed invariant one-form \(\beta\), then \(\mathrm d(\beta(\eta))=-\iota_\eta\mathrm d\beta=0\). The constant \(\beta(\eta)\) is zero on the axis, and thus everywhere. Finally \(\alpha(\eta)=O(r^{-1})\), while \(E\cdot\nu=O(r^{-2})\); the electromagnetic surface term is \(O(r^{-1})\) on the end spheres. This proves its disappearance in the limiting flux. The proof does not identify the bare gravitational flux on \(S\) with the flux at infinity. ◻

Proof. An invariant smooth scalar is even in \(y\). A smooth invariant horizontal vector field has an odd radial component and an even longitudinal component. Since \(X/y\) is smooth positive even, \(X*_hE_H\) and \(X*_hB_H\) have radial component \(O(y)\) and longitudinal component \(O(y^2)\) with the corresponding parities. Integration from the axis gives the first two assertions.

Smoothness of an invariant symmetric tensor gives an angular–axis component \(l_x=O(y)\) and an angular–radial component \(l_y=O(y^2)\). After multiplication by \(X^2\) and rotation by \(*_h\), \(\omega\) has radial component \(O(y^3)\) and longitudinal component \(O(y^4)\). Put \(\delta\psi=\psi-\psi_{\rm ax}\) and \(\delta\chi=\chi-\chi_{\rm ax}\). Then \[\mathrm d(z+\psi_{\rm ax}\chi-\chi_{\rm ax}\psi) =\omega-\delta\psi\,\mathrm d\chi +\delta\chi\,\mathrm d\psi.\] The last two terms have the same radial and longitudinal orders as \(\omega\). Integration proves the order-four assertion.

For the end estimates, perform the same argument in rescaled annuli. The leading electric field is the radial Coulomb field, so the leading angular derivative of \(\psi\) is \(Q\sin\vartheta\,\mathrm d\vartheta\). The remainder contributes a uniformly bounded function of order \(\sin^2\vartheta\) at a pole. The decaying invariant magnetic primitive gives the sharper estimate for \(\chi\): its normalized angular component vanishes linearly in transverse radius, and \(\chi=\alpha(\eta)\) vanishes quadratically. Finally \(X^2l=O(r^{-1}\sin^2\vartheta)\) as a horizontal covector, with the stronger radial and longitudinal polar orders just proved. Integration along a meridian of a coordinate sphere therefore bounds \(z\) and gives its order-two polar difference. For the stronger corrected estimate, fix a dyadic annulus of radius \(R\) and use coordinates scaled by \(R\), with signed transverse coordinate \(y\). The rescaled Cartesian tensor \(R^3K(R\,\cdot)\) has uniform \(C^3\) bounds and the same invariant tensor parity. The pullback of \(\omega=X^2*_hl\) therefore has transverse component \(O(y^3)\) and longitudinal component \(O(y^4)\), uniformly in \(R\): the factors from \(X^2\) and a covector pullback cancel the overall \(R^{-3}\) decay of \(K\). The first two potential estimates and their differentiated versions give these same orders for \(-(\psi-\psi_{\rm ax})\mathrm d\chi +(\chi-\chi_{\rm ax})\mathrm d\psi\). Integrating their sum with \(\omega\) from \(y=0\) gives \(\widetilde z=O(y^4)\) uniformly. On the unit-scale annulus \(y\) is comparable to \(\sin\vartheta\), proving Equation (24). Smooth polar coordinate changes with uniformly bounded transverse ratios preserve these estimates. Differentiating the defining equations supplies the radial decay and angular derivative bounds. The assumptions \(g-\delta=O_4(r^{-1})\), \(K=O_3(r^{-3})\) and \(E,B\) with three controlled derivatives give all the asserted rescaled bounds. ◻

Proof. First consider smoothness. In the normalized polar coframe \((\mathrm dy,\mathrm dx,y\mathrm d\phi)\), the angular–radial metric coefficient is even of order \(y^2\), and the angular–longitudinal coefficient is odd of order \(y\). The ratio \(X/y\) is smooth positive even. Taking the horizontal Schur complement of the angular coefficient therefore preserves all polar parities and the equality of the radial and angular diagonal coefficients at the axis. The same calculation for the tensor gives the smooth lift of \(k_0\). These statements apply uniformly on rescaled end annuli. Apparent factors \(y^{-1}\) in Cartesian differentiation are harmless: the Taylor terms permitted by the polar parities are smooth transverse Cartesian polynomials, and Taylor remainder estimates bound every remaining divided derivative. This also proves the required differentiated end decay. In particular \[\mathcal A_y=O(yr^{-3}),\qquad \mathcal A_x=O(r^{-2}),\qquad Xf_a=O(r^{-2})\] on the end. On compact axis tubes \(\mathcal A_y\) is smooth odd and \(\mathcal A_x\) smooth even, so \(f_a\) is odd of order \(y\). Smooth invariant vector fields have angular component odd of order \(y\), proving the stated parities of \(s\).

An invariant surface is generated by a meridional curve, and its area is \(2\pi\) times that curve’s length in \(X^2h\). This is the same for \(g\) and \(g_0\). First variation of this expression for arbitrary invariant normal speeds shows that its pointwise mean curvature is also unchanged. Its tangent space is generated by one horizontal unit vector and \(e\), so its tangential trace uses only \(K|_{H\times H}\) and \(K(e,e)\). This proves preservation of future expansion.

For the energy, use the end azimuth fixed in Lemma 3. Let \(\mathcal R\) be azimuthal reflection. The difference of \(g\) from its reflection average is precisely the angular cross tensor, which is odd under \(\mathcal R\). The linear ADM flux on a coordinate sphere is invariant under this Euclidean reflection, and hence is zero on that odd tensor. The difference of the reflection average from \(g_0\) is \(X^2\mathcal A\otimes\mathcal A\) in horizontal coordinate components. It has Cartesian order \(O_2(r^{-2})\), since it is quadratic in the normalized cross coefficients \(O(r^{-1})\). Its ADM flux is \(O(r^{-1})\). Thus the energies agree. The polar estimates above make the calculation uniform at both end poles.

It remains to prove the source identities, including their signs. The horizontal Koszul formulas, for invariant horizontal fields \(a,b\), are \[\begin{align*} (\nabla_a b)^e&=-\tfrac12X\mathrm d\mathcal A(a,b),& (\nabla_a e)^H=(\nabla_e a)^H &=\tfrac12X\mathrm d\mathcal A(a,\cdot)^\sharp,& \nabla_e e&=-\nabla_h\log X. \end{align*}\] Together with the base connection these give \[ \begin{gathered} R_{g_0}-R_g=\tfrac12X^2f_a^2,\qquad |K|_g^2-|k_0|_{g_0}^2=2|l|_h^2,\qquad \mathop{\mathrm{tr}}_{g_0}k_0=\mathop{\mathrm{tr}}_gK,\\ 8\pi(J_0)_i=8\pi(J_{\rm con})_i +X(\mathrm d\mathcal A)_{ij}l^j \quad(i\text{ horizontal}),\qquad J_0(e)=0. \end{gathered} \tag{27}\] For example the last vanishing also follows from the azimuthal reflection symmetry of \((g_0,k_0)\). The scalar constraint and \[\begin{align*} |\mathcal D_X|^2&=|E_H|^2+|B_H|^2+|l|^2,\\ (E\times B)_H&=E_e*_hB_H-B_e*_hE_H,\qquad (\mathrm d\mathcal A)_{ij}l^j=-f_a(*_hl)_i \end{align*}\] now give Equation (26) exactly.

For clarity, the normalized first two rows of \(\mathcal D_X\) need not individually be smooth three-dimensional covectors at the axis. Their radial coefficients are even and their longitudinal coefficients odd. Their squared norms, and their products with the corresponding odd \(s_i\), are smooth invariant scalars and covectors. The third row is a smooth horizontal covector: its radial coefficient is odd of order \(y\), and its longitudinal coefficient is even of order \(y^2\). Thus all source combinations in Equation (26) are smooth. Finally \(2|s\cdot\mathcal D_X|\le |s|^2+|\mathcal D_X|^2\) implies \(\mu_0\ge|J_0|_{g_0}\). ◻

Proof. Let \((\theta^0,\theta^1,\theta^2)\) be an oriented Lorentz orthonormal coframe with \(\theta^0(n)=1\), and put \(\theta^3=X(\mathrm d\phi+\mathbf A)\). The relevant signs are \[*_H\theta^0=-\theta^1\wedge\theta^2,\qquad *_H\theta^1=-\theta^0\wedge\theta^2,\qquad *_H\theta^2=\theta^0\wedge\theta^1.\] Set \(p=X^{-1}\mathrm d\chi\), \(t=X^{-1}\mathrm d\psi\), and \(v=X^{-2}\omega\). Then \[ F=p\wedge\theta^3+*_Ht,\qquad *_4F=-t\wedge\theta^3+*_Hp,\qquad \mathrm d\mathbf A=2X^{-1}*_Hv. \tag{30}\] Contraction with \(n\) and restriction to its spatial slice give exactly \(E^\flat=\iota_nF\) and \(B^\flat=*_g(F|_{T\Omega})\). In particular \(t(n)=-B_e\) and \(p(n)=E_e\). Writing \(\mathrm d\mathbf A=\theta^0\wedge a+f_a\theta^1\wedge\theta^2\), the mixed connection formula gives \(l=Xa/2\) for the positive future-normal convention. Consequently \[\omega|_{T\Omega}=X^2*_hl, \qquad X^{-2}\omega(n)=-Xf_a/2.\] This proves the restrictions in Equation (28).

For completeness, the integrability of the corrected twist in spacetime follows from the mixed Einstein equation. The mixed Ricci component is \[(\mathop{\mathrm{Ric}}_4)_{ie} =\frac1{2X^2}\nabla_H^j\big(X^3(\mathrm d\mathbf A)_{ij}\big) =X^{-2}\nabla_H^j(*_H\omega)_{ij}.\] Einstein–Maxwell gives it as \(-2(*_Ht)_{ij}p^j\). These identities are equivalent to \(\mathrm d\omega=2\mathrm d\psi\wedge\mathrm d\chi\), and hence to the closedness of \(\omega+\chi\mathrm d\psi-\psi\mathrm d\chi\). Thus \(z\) exists locally, with its constants fixed by the slice.

The horizontal parts of the Maxwell equations give \(\mathop{\mathrm{div}}_Ht=2\langle p,v\rangle_H\) and \(\mathop{\mathrm{div}}_Hp=-2\langle t,v\rangle_H\), while \(\mathrm d^2\mathbf A=0\) gives \(\mathop{\mathrm{div}}_H(X^{-1}v)=0\). For a covector \(\beta\), the conformal change in dimension three obeys \(\mathop{\mathrm{div}}_\gamma\beta=X^{-3}\mathop{\mathrm{div}}_H(X\beta)\). Therefore \[\begin{align*} \mathop{\mathrm{div}}_\gamma(X^{-4}\omega)&=0,\\ \mathop{\mathrm{div}}_\gamma(X^{-2}\mathrm d\psi) &=2X^{-4}\langle\mathrm d\chi,\omega\rangle_\gamma,\tag{31}\\ \mathop{\mathrm{div}}_\gamma(X^{-2}\mathrm d\chi) &=-2X^{-4}\langle\mathrm d\psi,\omega\rangle_\gamma. \end{align*}\]

The horizontal and fiber curvature contractions of the connection formulas are \[\begin{align*} (\mathop{\mathrm{Ric}}_4)_{ij} &=(\mathop{\mathrm{Ric}}_H)_{ij}-X^{-1}\nabla_i\nabla_jX -\tfrac12X^2(\mathrm d\mathbf A)_{ik}(\mathrm d\mathbf A)_j{}^k, \\ (\mathop{\mathrm{Ric}}_4)_{ee} &=-X^{-1}\Box_HX +\tfrac14X^2(\mathrm d\mathbf A)_{ij}(\mathrm d\mathbf A)^{ij}. \tag{32}\end{align*}\] One can see the coefficients directly before contraction: the horizontal sectional numerator has correction \(-3X^2\mathrm d\mathbf A(a,b)^2/4\), and the horizontal–fiber block is \(-X^{-1}\nabla_a\nabla_bX+ X^2(\mathrm d\mathbf A)_{ak}(\mathrm d\mathbf A)_b{}^k/4\). The Lorentz Hodge identity \[(*_Ht)_{ik}(*_Ht)_j{}^k=t_it_j-H_{ij}|t|_H^2\] shows that the two Einstein–Maxwell Ricci blocks in Equation [a:ricci-blocks] equal, respectively, \[2(p_ip_j+t_it_j)-H_{ij}(|p|_H^2+|t|_H^2), \qquad |p|_H^2+|t|_H^2.\] Thus the fiber equation reads \(X^{-1}\Box_HX=-|p|_H^2-|t|_H^2-2|v|_H^2\). Since \(u=\log X\) and \(\gamma=e^{2u}H\), this becomes \[ \Box_\gamma u =-e^{-2u}(|\mathrm d\psi|_\gamma^2+|\mathrm d\chi|_\gamma^2) -2e^{-4u}|\omega|_\gamma^2. \tag{33}\] The conformal Ricci formula \[\mathop{\mathrm{Ric}}_\gamma=\mathop{\mathrm{Ric}}_H-\mathop{\mathrm{Hess}}_Hu+\mathrm du\otimes\mathrm du -(\Box_Hu+|\mathrm du|_H^2)H\] then cancels the remaining scalar multiples of \(H\) and gives \[\mathop{\mathrm{Ric}}_\gamma=2(\mathrm du\otimes\mathrm du+p\otimes p+t\otimes t+v\otimes v) =2\Phi^*G.\] Equations [a:map-divergences] and (33) are exactly the four Euler–Lagrange equations of the target metric in Equation (21). This proves Equation (29).

Conversely, start with Equation (29), put \(H=X^{-2}\gamma\), and define \(p,t,v\) as above. The first equation in [a:map-divergences] makes \(2X^{-1}*_Hv\) closed, so it has a local connection primitive \(\mathbf A\). Define \(F\) by Equation (30). The remaining divergence equations and the identities \(\mathrm d^2\psi=\mathrm d^2\chi=0\) give both Maxwell equations. The identity \(\mathrm d\omega=2\mathrm d\psi\wedge\mathrm d\chi\) supplies the mixed Einstein equation. Reversing the horizontal and fiber calculations supplies the other Einstein equations.

The slice reconstruction is explicit. Its horizontal tensor is the second fundamental form of its slice in \(H\); the other components are \[ K(e,e)=n(\log X),\qquad X^2*_hK(e,\cdot)|_H=\omega|_{T\Omega},\qquad (X^{-1}\mathrm d\psi(n),X^{-1}\mathrm d\chi(n),X^{-2}\omega(n))=s. \tag{34}\] The spatial potential equations recover \(E_H,B_H\), and the first two velocities recover \(B_e,E_e\). The third recovers the spatial connection curvature. A chosen connection on a simply connected meridian is then fixed up to angular gauge. Finally the inverse constant duality rotation recovers the original ordered pair of electric and magnetic fields. ◻

Proof. A path of bounded \(G\)-length has bounded \(u\). The same length then controls the variations of \(\psi\) and \(\chi\). Since these coordinates are bounded, control of \(\mathrm dz+\psi\mathrm d\chi-\chi\mathrm d\psi\) also controls the variation of \(z\). A finite-length escaping path is therefore impossible, which proves completeness. The underlying manifold is \(\mathbb R^4\).

Let \(E_0,E_1,E_2,E_3\) be the frame dual to \((\mathrm du,e^{-u}\mathrm d\psi,e^{-u}\mathrm d\chi,e^{-2u}\omega)\). Its nonzero brackets are \[[E_0,E_1]=E_1,\quad [E_0,E_2]=E_2,\quad [E_0,E_3]=2E_3,\quad [E_1,E_2]=-2E_3.\] In the bivector order \((01,02,12,03,13,23)\), Koszul’s formula gives the curvature operator \[\begin{pmatrix} -1&0&0&0&0&-1\\ 0&-1&0&0&1&0\\ 0&0&-4&2&0&0\\ 0&0&2&-4&0&0\\ 0&1&0&0&-1&0\\ -1&0&0&0&0&-1 \end{pmatrix}.\] Its quadratic form on \((a,b,c,d,e,f)\) is \(-(a+f)^2-(b-e)^2-(c+d)^2-3(c-d)^2\), so it is nonpositive. The geodesic and convexity assertions follow from the Cartan–Hadamard theorem and the second variation formula. ◻

Proof. Substitute \(4M^2=R^2+2Q^2+C/R^2\) and \(a^2=J^2/M^2\). Clearing denominators gives the first two identities. Ordinary differentiation gives the third and also \[b'(R)= \frac{R^8+4Q^2R^6+6CR^4+4Q^2CR^2+C^2} {2R^2(R^4+2Q^2R^2+C)^{3/2}}>0.\] At the lower endpoint \(b\to0\) (also when \(C=0\), where the endpoint is \(R=0\)); as \(R\to\infty\), \(b\sim R/2\). ◻

Proof. The axial radius and quotient are \[ X_*^2=(W/P)\sin^2\vartheta, \qquad \gamma_*=-b^2\sinh^2\sigma\sin^2\vartheta\,\mathrm dt^2 +W\sin^2\vartheta (\mathrm d\sigma^2+\mathrm d\vartheta^2). \tag{38}\] Both Maxwell contractions in Equation (28) have zero time component. The same holds for \(*_H\mathrm d\mathbf A\), since \(\mathbf A=-a(2Mr-Q^2)\mathrm dt/W\) has curvature of the form \(\mathrm dt\wedge\) a spatial one-form. Thus the map is time independent.

An explicit check of all potential constants is useful. With \(x=\cos\vartheta\), they are \[\begin{align*} \psi_*&=-\frac{Q(r^2+a^2)x}{P},& \chi_*&=-\frac{Qar(1-x^2)}P,\\ z_*&=aMx(3-x^2) +\frac{a^3Mx(1-x^2)^2-aQ^2rx(1-x^2)}P.&& \tag{39}\end{align*}\] These expressions satisfy the contractions and corrected twist in Equation (28), by differentiation. In particular the axis values are \((-Q,0,2aM)\) and \((Q,0,-2aM)\). To check the angular sign directly, the shift is \[\beta^\phi=-2aM/r^3+O(r^{-4}).\] Stationarity and the positive convention \(K=\tfrac12\mathcal L_ng\) give \(K=-\mathcal L_\beta g/(2N)\) on the canonical slice. Hence \[K_{r\phi}=-3aM\sin^2\vartheta/r^2+O(r^{-3}), \qquad \frac1{8\pi}\int_{S_r}K(\nu,\eta)\,\mathrm dA\longrightarrow-aM.\] The leading term \(Q\mathrm dt/r\) of \(\mathfrak a_*\) gives \(F_{tr}=Q/r^2+O(r^{-3})\) and therefore positive electric charge \(Q\) with \(E=\iota_nF\). The magnetic flux is zero.

The horizon is \(r=M+b\), or \(\sigma=0\). There \(W=(r^2+a^2)^2=R^4\) by Equation (35); its area density is \(R^2\sin\vartheta\,\mathrm d\vartheta\mathrm d\phi\), proving the area and the boundary value of \(q_*\). In the Euclidean end radius \(\varrho=(b/2)e^\sigma\), one has \(r=\varrho+M+b^2/(4\varrho)\). Expanding the spatial metric in the associated Cartesian chart gives \[(g_*)_{ij}=(1+2M/\varrho)\delta_{ij}+O_4(\varrho^{-2}),\] and hence ADM energy \(M\). The shift and potential give \(K_*=O_3(\varrho^{-3})\), the electric Coulomb leading term, and \(B_*=O_3(\varrho^{-3})\), with uniform polar derivatives.

Put \(\rho_*=bw\). Equation (38) is static, with spatial metric \(e^{2q_*}(\mathrm d\sigma^2+\mathrm d\vartheta^2)\). The wave-map equation in Equation (29) is therefore \(\mathop{\mathrm{div}}_G(\rho_*\mathrm d\Phi_*)=0\), giving the first equation in Equation (37). Its spatial Ricci trace is \[-2e^{-2q_*}\Delta q_* -\rho_*^{-1}e^{-2q_*}\Delta\rho_* =2e^{-2q_*}|\partial\Phi_*|_G^2.\] Since \(\Delta\rho_*=0\), this proves the second equation.

Finally all the displayed spatial coefficients and potentials are smooth even functions of \(\sigma\) at zero, with the ordinary polar parity in \(\vartheta\). The apparent lapse division in the canonical \(K_*\) and fields is removable. Indeed the angular velocity \(\omega_*=a(2Mr-Q^2)/W\) is constant on the horizon: \(\omega_H=a/R^2\). Thus its \(\vartheta\) derivative is \(O(\sigma^2)\) and its \(\sigma\) derivative is \(O(\sigma)\), while the lapse is \(\sigma\) times a smooth positive even function. The electrostatic potential on \(\partial_t+\omega_H\partial_\phi\) is also constant on the horizon; its nonconstant remainder is \(O(\sigma^2)\). These orders give smooth canonical tensor and field components after division by the lapse. The metric has no conical defect at the axes, and the explicit potentials give precisely the order-two and corrected order-four axis behavior established earlier. For a two-sided extension across \(\sigma=0\), the stationary lapse is taken with its smooth sign, \(b\sinh\sigma\sqrt{P/W}\); the stationary Killing field changes time orientation across the bridge. This supplies a smooth future normal and tensor, whereas using the absolute value of that lapse would only reflect the chosen stationary time orientation. This proves all the regularity assertions. ◻

Proof. All losses below can be made arbitrarily small. We first remove the extra puncture tails, while retaining the original weight elsewhere. Take pairwise disjoint puncture neighborhoods and modify the weight only where \(t>T\). A positive bounded lower extension exists there: in the main strip, the puncture comparison implies \[\frac{B}{\sin\alpha}\asymp \frac{\abs{Z}^2}{\abs{Z-a_j}^2}.\] Indeed \(\sin\beta=\operatorname{Im}Z/\abs{Z-a_j}\) and \(\abs{\mathrm d\log(Z-a_j)}=\abs{Z}\abs{Z-a_j}^{-1} \abs{\mathrm d\log Z}\). Thus a small positive smooth quotient can replace the divergent quotient near \(a_j\), with the transition wholly in \(t>T\).

These modifications cannot decrease the crossing infimum by a fixed positive amount as \(T\to\infty\). To prove this, suppose that modified crossings of length at most a fixed \(M\) did produce such a loss. Enlarge each original side by all the modified tails attached to that side. Between the last encounter with the initial enlarged side before the first encounter with the terminal enlarged side, keep the intervening path. It meets no modified tail in its interior. If an endpoint is in a tail, this path must traverse the unchanged collar \(T/2<t<T\) to leave its puncture neighborhood. On that traversal, \[M\ge c_j\int\sin\beta\,\abs{\mathrm dt} \ge \frac{c_jT}{2}\min\sin\beta.\] At a point attaining, or arbitrarily approaching, this minimum, attach the path to the nearer local ray by a constant-\(t\) angular interval. Its original length is at most \[C_j(1-\abs{\cos\beta})\le C_j\sin^2\beta \le C_j\left(\frac{2M}{c_jT}\right)^2.\] Both local rays at \(a_j\) lie on the same original global side. Attaching at the two ends of the retained path therefore produces an original opposite-side crossing with only \(O(T^{-2})\) additional length. The finitely many disjoint collars make this estimate uniform. This contradicts a fixed loss. Taking \(M>L\) suffices, since longer paths cannot violate any proposed lower bound below \(L\).

At the negative end, uniform convergence permits replacement by a smooth even profile below \(b_-\), with arbitrarily small relative loss. A transition can be made within a region where the original quotient is uniformly close to \(b_-\). At the positive end, fix a small \(\delta>0\) and take \(V\) large enough that \[(1-\delta)c_+e^{2v}\le B/\sin\alpha \le(1+\delta)c_+e^{2v}\qquad(v\ge V-1).\] Blend downward to \((1-\delta)c_+e^{2v}\) before \(v=V\), and thereafter use \((1-\delta)c_+e^{2\min(v,V)}\). Radial projection \((v,\alpha)\mapsto(\min(v,V),\alpha)\) converts a path for this lower weight into a path for the preexisting weight with length increased by at most \((1+\delta)/(1-\delta)\). Below \(V\) this follows from the same ratio comparison on the transition; above \(V\), projection deletes the radial component, and the angular weight at \(V\) is at most that ratio times the saturated weight. Projection preserves the two sides. Consequently saturation loses only this arbitrarily small factor in the crossing lower bound.

The resulting positive continuous quotient is bounded above and below, and is constant in \(v\) on both far ends. Approximate it uniformly from below by a smooth quotient, even at the sides and still product on the ends. Such approximation follows by even reflection, convolution on the compact remaining rectangle, and a small downward adjustment; the positive lower bound converts uniform error to relative error. The product profiles can be approximated in the same way before extending them along the tails. The accumulated relative and additive losses can be chosen smaller than \(L-\ell\). This proves the lemma. ◻

Proof. Choose a lower weight with crossing infimum at least \(\ell>L'\), leaving strict room for the losses that follow. Let \(d_+\) denote its distance to the side \(\alpha=0\). The distance inequality gives \(\abs{\mathrm dd_+}\le W\) almost everywhere. Since \(m_0\le b_0\le M_0\), angular intervals give \[d_+(v,\alpha)\le M_0(1-\cos\alpha),\qquad d_+(v,\alpha)\ge\ell-M_0(1+\cos\alpha).\] The second inequality follows by appending an angular interval from the point to the opposite side to an almost minimizing path for \(d_+\). For a small \(h>0\), truncate \(d_+\) to \([h,\ell-h]\) and put \[\varphi_0 =1-2\min\left\{1,\max\left\{0, \frac{d_+-h}{\ell-2h}\right\}\right\}.\] Thus \(\abs{\mathrm d\varphi_0}\le2W/(\ell-2h)\), and \(\varphi_0\) is identically \(1\) and \(-1\) on full, uniform collars of the respective sides. Take \(h\) small enough to retain strict slack above \(L'\).

Clamp \(v\) to a large closed interval with endpoints inside the product tails. This projection is length nonincreasing for \(W\), so composition preserves the gradient bound and makes \(\varphi_0\) independent of \(v\) on the far tails. Smooth after even reflection at the sides. The constant side collars remain constant; elsewhere \(W\) has a positive lower bound on a compact rectangle, so convolution and a small slack loss preserve the weighted gradient bound. This produces a smooth function. A sufficiently small convex combination with \(\cos\alpha\) then gives strict interior values and \[\partial_{\alpha\alpha}\varphi(v,0)<0, \qquad \partial_{\alpha\alpha}\varphi(v,\pi)>0.\] Its gradient cost fits within the slack, because \(\sin\alpha\le W/m_0\).

On a product end write \(W=b_\pm(\alpha)\sin\alpha\) and set \[I_\pm=\int_0^\pi b_\pm(s)\sin s\,\mathrm ds, \qquad \varphi_\pm(\alpha) =1-\frac{2}{I_\pm}\int_0^\alpha b_\pm(s)\sin s\,\mathrm ds.\] The vertical crossing on that end gives \(I_\pm\ge\ell\). Both the smoothed function and \(\varphi_\pm\) have the same endpoint values and even reflection, so their difference is \(O(\sin^2\alpha)\), uniformly on the end. Interpolate to \(\varphi_\pm\) sufficiently slowly in \(v\). The angular derivative is bounded by the convex combination of the preceding bounds. The new \(v\) derivative is bounded by \(C\abs{\chi'}\sin^2\alpha\), which is arbitrarily small relative to \(W\) when the interpolation interval is long. Thus, for some \(L_0>L'\), \[ \abs{\mathrm d\varphi}\le\frac{2}{L_0}W. \tag{42}\] The angular derivative is strictly negative on the far ends and in small uniform side collars. A compactly supported arbitrarily small Morse perturbation away from those collars leaves these properties intact and leaves only finitely many critical points. We retain the notation \(\varphi\) and a slightly smaller \(L_0>L'\).

We next replace its level sets by a foliation without losing the estimate. Delete small open intervals about its finitely many critical values. In a closed remaining regular value band, intersect a level with \([-V,V]\times[0,\pi]\), taking \(V\) in the common product tails. There is exactly one level point on each vertical edge and no point on a horizontal edge. A compact regular level is a union of circles and intervals; its two boundary points must therefore be joined by one interval component. This is the unique essential component.

For an entire closed regular band, a vector field satisfying \(\mathrm d\varphi(V_\varphi)=1\), chosen tangent to the vertical edges, flows these essential components smoothly into one embedded rectangle. It has precisely the two extreme essential arcs and the appropriate vertical-edge intervals as its boundary. The Jordan separation theorem therefore identifies its image with the full region between the two arcs. In particular, discarding the other, circular level components creates no holes in this essential band. Essential bands at distinct values are disjoint and ordered, since their order is fixed at both vertical ends and disjoint crossing arcs cannot exchange that order. Near \(1\) and \(-1\) keep the monotone side collars: outside smaller such collars the values stay uniformly away from the extrema, by compactness and the product-end behavior.

These finitely many essential bands can be completed to a foliation of the whole strip by proper arcs. Here is the extension with its boundary conditions. Slightly shrink each good band, retaining its level coordinate on a neighborhood of the resulting closed band. Between two consecutive bands, their smooth boundary arcs and the two vertical edges bound a disk. Prescribe product parametrizations on collars of all four edges, using the original label where it is already fixed and the existing angular labels on the vertical ends. They agree at the corners because the ends are product. Shrink the collars so that nonadjacent ones are disjoint and round within the compatible corner charts. Smooth Schoenflies charts reduce the extension to a prescribed orientation-preserving collar map of a disk. Here the relative extension can be described explicitly. In polar coordinates its boundary derivative is triangular, with positive diagonal entries. Interpolation to the radial product map is therefore an isotopy on a sufficiently thin collar; extend its generating vector field with a cutoff over the disk. For the resulting boundary map, choose an increasing lift \(h\) with \(h(\theta+2\pi)=h(\theta)+2\pi\). The map \[(r,\theta)\longmapsto \bigl(r,\theta+\chi(r)(h(\theta)-\theta)\bigr),\] where \(0\le\chi\le1\), \(\chi=1\) near the boundary, and \(\chi=0\) near the center, is a smooth disk diffeomorphism: its angular derivative is \(1-\chi+\chi h'>0\), and it is the identity near the center. Undoing the extended collar isotopy gives an extension agreeing with the entire prescribed smaller collar. Thus there is no additional corner or orientation condition to impose. Extending by the product ends gives a smooth value function \(\widehat\varphi\), with no interior critical points, agreeing with \(\varphi\) on the smaller good bands and on the side collars. It has the same endpoint values and reflection parity. The labels in every intervening strip lie solely in its assigned complementary value interval. Thus discarded circles do not reappear as extra components of a good label.

There is no a priori gradient estimate in an intervening strip, but there are only finitely many of them, they avoid the side collars, and they are product at infinity. Hence \(\abs{\mathrm d\widehat\varphi}/W\) has a finite bound there, say \(M\). The total width \(\delta\) of their label intervals, including small transition intervals within good bands, can be chosen arbitrarily small. Choose a smooth positive function \(r\) of the label, equal to \(1\) away from those intervals, at most \(1\) everywhere, and at most \(\varepsilon\) on every bad label interval; arrange its transitions inside good bands. Put \[F(t)=-1+\frac{2}{\int_{-1}^1r(s)\,\mathrm ds} \int_{-1}^t r(s)\,\mathrm ds.\] The normalization factor is at most \(2/(2-\delta)\). On good bands the old gradient estimate is multiplied by at most this factor. On bad bands it is bounded by this factor times \(\varepsilon M\). First choose \(\delta\) small, and then \(\varepsilon\) small. The function \(F\circ\widehat\varphi\) satisfies Equation (41) for some \(L_1>L'\). It has no interior critical points. Since \(F'>0\) at the endpoints, its side jets remain nondegenerate and even. We again call this function \(\varphi\).

Set \(\Theta=\arccos\varphi\). In the interior this is a submersion. At \(\alpha=0\), the even expansion has the form \(1-\varphi=\alpha^2 a(v,\alpha^2)\) with \(a(v,0)>0\); smooth square-root division shows that \(\Theta\) is \(\alpha\) times a positive smooth even function. The same argument applies to \(\pi-\Theta\) at the other side. Thus \(\Theta\) has precisely the stated reflection and nonvanishing boundary derivative.

For use in the analytic argument we also record an isotopy. Integrate \(D\Theta/\abs{D\Theta}^2\) from the lower edge, with \(\Theta\) as time. The vector field is vertical on the product ends and has bounded coefficients in the remaining compact region. Its flow gives a strip diffeomorphism \(D(v_0,\theta)\) with \(\Theta(D(v_0,\theta))=\theta\). Reflection reverses the flow parameter and gives ordinary even/odd parity for the two components of \(D\). In particular \(D\) is product on uniform far ends.

Such a diffeomorphism is smoothly isotopic to the identity within this class. First straighten its two end angular diffeomorphisms by convex angular interpolation, whose derivative stays positive, and extend these collar motions by their generating vector fields. Next straighten the parametrizations of the horizontal sides, if necessary, by convex interpolation of their increasing longitudinal parametrizations; these motions are compactly supported in \(v\). After this, the derivative on each side is diagonal with positive entries by reflection. Linear interpolation to the identity in a sufficiently thin collar is consequently a local collar isotopy. Extend its generating field with a symmetric cutoff. The remaining map fixes a neighborhood of the entire boundary of a compact disk. After choosing a smaller fixed boundary collar, the smooth disk isotopy theorem (Smale 1959, Theorem 4) joins it to the identity through orientation-preserving diffeomorphisms fixing that collar. Reattaching the fixed collars and ends, and taking inverse coordinate maps, gives the required family of angular submersions. All these uses of disk topology concern smooth parametrizations of planar disks; no metric estimate is required during the isotopy. ◻

Proof. Let \(\Theta_s\), \(0\le s\le1\), be the isotopy, with \(\Theta_0(v,\alpha)=\alpha\) and \(\Theta_1=\Theta\). For fixed \((s,\vartheta)\), the allowed values \(u=x+iy\) form the line \[\Gamma_{s,\vartheta} =\{x+iy:\Theta_s(\log k+x,\vartheta+y)=\vartheta\}.\] It is a smooth proper embedded line with horizontal ends. These lines vary smoothly, periodically in \(\vartheta\), with \(\Gamma_{s,-\vartheta}=\overline{\Gamma_{s,\vartheta}}\). At \(\vartheta=0,\pi\) the line is the real axis. Their heights and the locations at which they become horizontal have uniform bounds for \(s\in[0,1]\) and \(\vartheta\) modulo \(2\pi\).

Orient each line from its left end to its right end. Its tangent phase has a smooth real lift \(\beta\), obtained by transporting the initial phase zero through the strip diffeomorphisms of the isotopy. It is periodic in \(\vartheta\) and odd under reflection. To check the potential integer ambiguity, at the left horizontal end its value is zero. At the right horizontal end its value is an integer multiple of \(2\pi\); that integer is continuous in the isotopy parameter and is zero for \(s=0\), so remains zero. The same uniqueness of the lift proves periodicity and reflection, with value zero on the straight boundary leaves. On compact sets of positions, the line data, the phase, and all their derivatives are uniformly bounded.

Proof of Lemma 11. Choose the angular calibration with \(L_1>L'\) and solve the analytic boundary problem at the prescribed \(k\). On the upper inner edge, \[\varphi\bigl(\log\abs{f(e^{i\vartheta})}, \arg f(e^{i\vartheta})\bigr)=\cos\vartheta.\] Consequently Equation (41) gives \[\sin\vartheta \le \frac{2}{L_1} W\bigl(\log\abs f,\arg f\bigr) \abs{\frac{\mathrm d}{\mathrm d\vartheta}\log f(e^{i\vartheta})} \le \frac{2}{L_1}\frac{\mathrm d\ell}{\mathrm d\vartheta}.\] This already proves Equation (40) with strict slack. If an endpoint coincides with an additional puncture, translate \(f\) by a sufficiently small real constant. This preserves symmetry, univalence, and the leading coefficient \(k\), and preserves enclosure of zero because the untranslated Jordan curve has positive distance from zero.

The slack survives such a translation, including at the endpoints. Indeed, in \(Z\) coordinates the smooth lower length density is \[\lambda_0(Z) =\frac{b_0(\log\abs Z,\arg Z)\operatorname{Im}Z}{\abs Z^2}.\] Away from zero it is smooth with odd reflection and has a positive simple zero at the real axis. For the translated smooth symmetric curve, the quotient \[\frac{\lambda_0(f(e^{i\vartheta})+t) \abs{\partial_\vartheta f(e^{i\vartheta})}} {\sin\vartheta}\] extends continuously to \(\vartheta=0,\pi\) and depends continuously on the small real translation \(t\). At \(t=0\) it is at least \(L_1/2\). Compactness of \([0,\pi]\) therefore preserves the bound \(L'/2\) for all sufficiently small \(t\). Avoiding the finitely many translations that put either endpoint at an \(a_j\) makes the cut miss every extra puncture. Its interior already lies strictly in the upper half-plane. The original density dominates this lower density throughout, so the required estimate follows for \(B\). The statement about remaining marked side points follows from symmetry and exterior univalence. ◻

Proof. Put \(C=Q^4+4J^2\). The model parameter relations give \[M=\frac12\sqrt{R^2+2Q^2+C/R^2},\qquad b=\frac{R^4-C}{4MR^2},\qquad M'(R)=\frac bR.\] Consequently \[ \frac{\mathrm d}{\mathrm dR}\left(M+\frac b2\log\frac{L}{2R^2}\right) =\frac{b'(R)}2\log\frac{L}{2R^2}. \tag{68}\] Here \(b'(R)>0\) by Lemma 9. If \(L/2>\sqrt C\), Equation (68) shows that the comparison expression increases up to \(R^2=L/2\) and decreases afterward. At that point its logarithmic term vanishes, proving (67). If \(L/2=\sqrt C\), use models with \(R^2\downarrow L/2\); then \(b\downarrow0\) and the logarithmic correction tends to zero. The limiting argument is numerical and uses only strictly subextremal comparison models. ◻

Proof. First show that every exterior support of a small parallel set \(D_z=\{x:\mathop{\mathrm{dist}}(x,D)\le z\}\) has expansion at most \(c+Cz\), with \(C\) depending on the compact geometry and \(Q\), but not on its curvature. Let such a support touch at \(x\), and take a shortest segment of length \(z\) from \(y\in D\) to \(x\). Choose a local unit vector field \(V\) agreeing at \(y\) with its initial tangent, and set \(F(y')=\exp_{y'}(zV(y'))\). Prescribe the first derivative of \(V\) so that \(\mathrm dF_y\) is exactly parallel transport along the segment. For each initial vector this is the Jacobi boundary-value problem with that vector and its parallel translate as the two endpoint values. Rescaling the Jacobi equation shows that the required initial derivative is \(O(z)\) times the initial norm. Its component along the segment is zero, since the parallel component of the Jacobi field is constant. Thus it is a permissible first jet of a unit field. More explicitly, if \(B_i=\nabla_iV\), the unit-length constraint on its symmetric second jet is \(\langle\operatorname{sym}\nabla^2_{ij}V,V\rangle =-\langle B_i,B_j\rangle\); its tangential symmetric part can be chosen bounded, and the antisymmetric part is fixed by the bounded curvature commutator. In a smooth orthonormal frame, normalizing a vector-valued polynomial with the prescribed value and first jet realizes precisely this compatibility. Thus \(V\) has bounded second derivatives independently of the support. Since \(F\) is the identity at \(z=0\), smoothness of the exponential map now gives \(|\nabla\mathrm dF|=O(z)\).

If \(\phi\) defines the support at \(x\), then \(\phi\circ F\) supports \(D\) at \(y\): every \(y'\in D\) maps a distance at most \(z\) from \(D\). The tangent ball at \(x\) aligns the support normal with the final segment tangent. Since \(\mathrm dF_y\) is exactly parallel transport, comparison of the two tangential Hessian traces after gradient normalization has error only \(O(z)\), coming from \(\nabla\mathrm dF\); there is no uncontrolled multiple of \(\nabla^2\phi\). Normal transport and variation of \(Q\) contribute another \(O(z)\). This proves the claimed parallel-set bound.

Choose \(0<a<b\) so small that \(c+Cb<c'\). Smooth approximation of the distance followed by a regular-value choice gives a smooth enclosure containing \(D_a\) and contained in \(D_b\); fill any pockets that meet no designated outer face. Let \(\eta\ge0\) be one on its boundary and supported where \(\mathop{\mathrm{dist}}(\cdot,D)<b\). Replacing \(Q\) temporarily by \[Q-\frac12(c'+A_0\eta)g\] makes this enclosure a strict inner barrier when \(A_0\) is sufficiently large. The designated outer barriers stay positive. Apply the strict barrier theorem to each complementary component meeting a designated outer face; every such component has both barriers, by connectedness and the strict enclosure. The resulting marginal surfaces together bound a region containing \(D_a\).

If the minimum distance \(z\) of that boundary from \(D\) were at most \(b\), its boundary at a minimizing point would be an exterior support of \(D_z\). Indeed a path from \(D\) of shorter length cannot cross that boundary before reaching it. Its expansion for the original \(Q\) is \(c'+A_0\eta\ge c'\), contradicting the strict parallel-set bound. Thus the boundary lies beyond \(D_b\), where \(\eta=0\), and has original expansion \(c'\). This proves the enclosure assertion without any regularity hypothesis on \(D\) or any support-curvature bound. ◻

Proof. Consider the auxiliary stationary Lorentz metric \(-l^2(\mathrm db-\mathrm df)^2+\bar g\). Its \(b\)-slice metric is \(g\), its lapse is \(U=l\sqrt D\), its shift is \(\beta=Uw\), and its slice tensor is \[k_s=-U^{-1}\operatorname{sym}\nabla\beta =-H^f-p(\mathrm dt\otimes w^\flat+w^\flat\otimes\mathrm dt).\] Set \(\mathfrak B(A,B)=A:B-(\mathop{\mathrm{tr}}A)(\mathop{\mathrm{tr}}B)\), \(P_A=A-(\mathop{\mathrm{tr}}A)g\), and \(Q_1=K-k_s\). The Gauss and normal-expansion identities, with stationarity, \(U\mathop{\mathrm{tr}}k_s=-\mathop{\mathrm{div}}\beta\), and, for the future unit normal \(n_{\rm L}\), \[\operatorname{Ric}_{\rm Lor}(n_{\rm L},n_{\rm L}) =-n_{\rm L}(\mathop{\mathrm{tr}}k_s)-|k_s|^2+U^{-1}\Delta U,\] give the two expressions for the spacetime scalar density \[ U\bigl(R_g+\mathfrak B(k_s,k_s)\bigr) -2\mathop{\mathrm{div}}\bigl(\nabla U+(\mathop{\mathrm{tr}}k_s)\beta\bigr) =U\bigl(R_{\bar g}-2l^{-1}\bar\Delta l\bigr). \tag{95}\] The right side is the static warped-product expression in the coordinate \(b-f\). Moreover, \[\bar\Delta l=\frac lU\mathop{\mathrm{div}}\left(\frac UlA_d\nabla l\right),\qquad \nabla U-\frac UlA_d\nabla l=-k_s(\beta,\cdot).\] Substitute the constraint equations and \(P_K:\nabla\beta=-U\mathfrak B(K,k_s)\) into Equation (95). The result is \[ UR_{\bar g} =U\left[16\pi(\mu+J_0(w))+\mathfrak B(Q_1,Q_1)\right] -2\mathop{\mathrm{div}}(UP_{Q_1}w). \tag{96}\] The conformal terms in \(\frac12e^{4t}R_{\hat g}\) are \(-4\bar\Delta t-4|\mathrm dt|_{A_d}^2\). Differentiating Equation (84) gives \[\begin{gather*} 4\mathrm dZ=(4+pv)\mathrm dt+aH^f(e,\cdot),\tag{97}\\ V/u=P_{Q_1}w+4A_d\nabla t+wF,\qquad \mathop{\mathrm{div}}(uw)/u=F+(1-\chi)q-\mathop{\mathrm{tr}}K+2(p+1)w(t). \tag{98}\end{gather*}\] Combining these formulas leaves the quadratic tensor expression \[\begin{align*} \mathcal T={}&\tfrac12\mathfrak B(Q_1,Q_1) +2P_{Q_1}(w,\nabla t)+4(p+1)|\mathrm dt|_{A_d}^2\\ &+F\bigl[F+(1-\chi)q+2(p+1)w(t)\bigr]. \tag{99}\end{align*}\] This expression is smooth at \(w=0\); for example \((1-\chi)q=\frac{p+4}{4+pv}S_1(w,w)\). Expanding with \(\mathop{\mathrm{tr}}T=F-\chi q\) gives Equations (89) and [d:quadratic]. Completing the mixed and normal squares gives Equation [d:squares], hence Equation (92). When \(\mathrm dt=0\), the normal quadratic form in \((\sqrt\chi q,F)\) has eigenvalues \(1\) and \((3-\chi)/4\), which lie between \(1/2\) and \(1\), proving Equation (93). Finally, at a constant-\(f\) face, \((\mathop{\mathrm{Hess}}f)|_{TB}=(\partial_\nu f)\mathrm{II}_B\). Substituting this into the trace equation gives the first boundary identity; the graph and conformal mean-curvature formulas give the second. ◻

Proof. At an inner boundary contact in the first three stages, \(j(t)=0\). Equations (103) and (94) therefore give \(4\partial_\nu t=-H-N_B<0\), incompatible with a minimum when \(\nu\) points into the domain. The preceding outer estimate excludes the outer face.

Suppose there is an interior contact. Then \(\mathrm dt=0\) and \(\mathop{\mathrm{Hess}}t\ge0\). The graph Gauss equation in the Riemannian warped product \(g+l^2\mathrm db^2\) gives \[ \frac12e^{4t}R_{\hat g} \le\frac12R_g-v\mathop{\mathrm{Ric}}_g(e,e)+\mathcal S(H^f), \tag{108}\] where, for a symmetric tensor \(A\), \[\mathcal S(A)=\frac14(\mathop{\mathrm{tr}}_\perp A)^2 -\frac12|A_{\perp\perp}^{\rm tf}|^2 +dA_{ee}\mathop{\mathrm{tr}}_\perp A-d|A_{e\perp}|^2.\] In fact \(l^{-1}\mathop{\mathrm{Hess}}l=p\mathop{\mathrm{Hess}}t\) at contact, so the warping term \(-v\mathop{\mathrm{tr}}_\perp(l^{-1}\mathop{\mathrm{Hess}}l)\) and the conformal term \(-4\bar\Delta t\) have favorable signs; \(H^f\) is the graph second form there, up to the immaterial choice of normal.

Directly from Equation [d:quadratic], at \(\mathrm dt=0\), \[ \mathcal T-\mathcal S(S_1) =|T^{\rm tf}|^2+(1+d)|M|^2 +\chi(1+d-\chi/2)q^2-dFq+\frac12F^2. \tag{109}\] Use \(\chi\le d\) and \(|dFq|\le\chi q^2/4+(d^2/\chi)F^2\) together with \(d/\chi\le1+n/4\) and Equation (93). This gives \[\mathcal T-\mathcal S(S_1) \ge\tfrac12\mathcal T-(1+n/4)F^2.\] Replacing \(S_1\) by \(H^f=S_1-K\) introduces only products involving transverse components and \(dq\). Young’s inequality and Equation (93) absorb an arbitrarily small fixed multiple of \(\mathcal T\), at polynomial cost in \(n\) for \(|K|^2\). Thus, for a fixed \(c_*>0\) independent of \(n\), \[ \mathcal T-\mathcal S(H^f) \ge c_*\mathcal T-\Pi_n(F^2+|K|^2). \tag{110}\] When \(K=0\), improve the last error to \(\Pi_n vF^2\). To see this, if \((1+p)v\) is sufficiently small, \(d\) and \(\chi\) are uniformly close to one and the normal quadratic form in Equation (109) is uniformly positive definite; its value at \(d=\chi=1\) is \(\frac32q^2-Fq+\frac12F^2\). Otherwise \(v\ge c/(1+n)\), so the previous error is bounded by a new polynomial times \(vF^2\).

At contact, differentiating \(w\) multiplies the vector slot of \(H^f\) by \(A_d\), and hence \(|\nabla w|\le\Pi_n(|K|+\sqrt{\mathcal T})\). View \(F\) as a smooth function of \((x,w,h,p)\). Its explicit \(x,w\) derivatives have compact support and polynomial bounds on the range in Equation (106); also \(\nabla p=0\) at contact. Since \(F_h\ge1\), for any fixed \(\eta_0>0\), \[ w(F)\ge\tau_0a\sigma-\eta_0\mathcal T -\Pi_n\mathbf1_{\rm comp}. \tag{111}\] Here \(\mathbf1_{\rm comp}\) denotes the indicator of a fixed sufficiently large compact set; it can equally be replaced by a fixed compactly supported majorant. The difference between the physical flux and \(V_\lambda\) costs at most \(\eta_0\mathcal T+\Pi_n\mathbf1_{\rm comp}\) after taking its divergence and dividing by \(u\). Indeed differentiate \(uA_\chi K(w,\cdot)\) using the preceding estimate. The remaining factor \(\nabla\log u=H^f(w,\cdot)\) contracts against \(A_\chi K(w,\cdot)\) and involves only \(\chi H^f_{ee}\) and \(H^f_{e\perp}\), which are controlled by Equation (93).

Compare Equations (89) and (108), and use Equations (110) and (111). Outside the fixed compact set, \(K=0\), \(F=h=O(r^{-\gamma})\), and \(\mathop{\mathrm{Ric}}_g=O(r^{-3})\). Choose \(\eta_0\) and then \(\delta_0\) sufficiently small compared with \(c_*\). The resulting lower bound is \[ u^{-1}\mathop{\mathrm{div}}V_\lambda \ge2\delta_0\mathcal T+\tau_0a\sigma -\Pi_n\bigl[\mathbf1_{\rm comp}+v(\rho_0+\rho_1)\bigr]. \tag{112}\] Choose \(C_n\) so that \(\mathcal P\) dominates the last error plus \(2\rho\) everywhere. This requires only polynomial growth, since \(\rho_0\) is positive on the compact set. At the floor \(m_0=1\), and subtracting the prescribed source from this geometric lower bound gives at least \[\delta_0\mathcal T+(1-\delta_0)\tau_0a\sigma+\rho>0,\] contradicting Equation (102).

In the fourth stage, \(f=0\), \(t=Z\), and \(v=0\). As long as the common scale factor is positive, the inner floor-contact derivative is that factor times \((-H-N_B)/4<0\). At an interior floor contact, \(\mathcal T\) is compactly bounded by \(K\); enlarge \(C_n\) so that \(\delta_0\mathcal T+\rho-\mathcal P<0\) there. The divergence of \(4u\nabla t\) is nonnegative at a minimum, giving a contradiction. At scale zero, the unique mixed-problem solution is \(Z=0\). This also cannot meet the floor. ◻

Proof. The quantifier sufficiently large truncations allows a threshold \(R_0(n)\) depending on \(n\). If the first conclusion fails, then for every \(\eta'>0\) and every \(n_0\) there is an \(n\ge n_0\) with failures at arbitrarily large \(R\). Choose successively \(\eta'=1/j\), \(n_j\ge\max\{j,n_{j-1}+1\}\) and \(R_j\ge\max\{j,R_{\rm floor}(n_j)\}\) among those failures, with \(R_{\rm floor}(n)\) the outer threshold in Lemma 22. There are points \(x_j\) in the fixed compact set with \(h_j(x_j)\le b_-+1/j\). Pass to \(x_j\to x_*\) and take the lower spatial relaxed limit \(\underline h\), defined by infima over tail indices and shrinking balls, followed by the lower semicontinuous envelope. Near a white face first extend each \(h\) continuously by the constant \(b_+\) behind the face. This does not assert continuity of the relaxed limit. By construction, \(\underline h(x_*)\le b_-\).

Away from black faces, every smooth lower test \(\phi\) for \(\underline h\) at a value sufficiently near or below \(b_-\), with \(\nabla\phi\ne0\), has oriented level expansion for \(K\) at most \(\underline h+C_b\). Indeed make the touching strict by a fourth-order perturbation and transfer it to local minima of \(h-\phi\) along a subsequence. The contacts are interior, because white boundary values are higher. There \(\nabla f=\tau_0^{-1}\nabla\phi\), so \[d,\chi\longrightarrow0,\qquad \frac{l}{\tau_0\sqrt D}\longrightarrow\frac1{|\nabla\phi|}.\] The lower Hessian comparison in the trace equation therefore gives the level expansion bound. The second-stage correction has \(D_0\le0\) at these low values, so it has the favorable sign.

We pass from this lower-test statement to exterior supports of closed sublevels. Fix \(b_-<k<k'\) sufficiently close to \(b_-\) and put \(D_k=\{\underline h\le k\}\). Apply the increasing reparametrizations \[r_j(s)=\left(1+\exp[-j^2(s-k-j^{-1})]\right)^{-1}.\] Their lower relaxed limit on \(\underline h\) is the function \(u_k\) equal to zero on \(D_k\) and one off \(D_k\). At a point outside the closed set, lower semicontinuity provides a local positive gap from \(k\); on the set itself the unchanged point gives limiting value zero. Let \(\phi\) define an exterior support at \(x_0\), with \(\phi(x_0)=0\), \(\nabla\phi(x_0)\ne0\), and \(D_k\) locally on its nonpositive side. Choose a small closed ball with \(|\phi|<1/4\), and set \(\phi_\eta=\phi-\eta|x-x_0|^4\), where \(\eta>0\). Then \(u_k-\phi_\eta\) has its unique minimum zero at \(x_0\) and a positive gap on the ball’s boundary. Minimize the lower semicontinuous function \(r_j(\underline h)-\phi_\eta\) on this ball. Because \(r_j(\underline h(x_0))\le r_j(k)\to0\), the lower relaxed limit property gives interior minimizers \(y_j\to x_0\), with minimum values \(c_j\to0\). Their transformed contact values \(a_j=\phi_\eta(y_j)+c_j=r_j(\underline h(y_j))\) tend to zero and, for each finite \(j\), lie strictly between zero and one.

On a sufficiently small neighborhood of \(y_j\) the smooth function \(\psi_j=r_j^{-1}(\phi_\eta+c_j)\) is therefore a lower test for \(\underline h\). The inverse is single-valued and strictly increasing; moreover \(\nabla\phi_\eta(y_j)\to\nabla\phi(x_0)\ne0\), so \(\nabla\psi_j(y_j)\ne0\) for large \(j\). Since \(r_j(k')\to1\), one also has \(\underline h(y_j)<k'\) for large \(j\). Increasing reparametrization preserves the gradient direction and the tangential Hessian divided by gradient length, regardless of the size of its derivatives. Apply the already proved lower-test inequality to each fixed \(\psi_j\), and then let \(j\to\infty\). The quartic perturbation has zero first and second jets at \(x_0\). Thus every exterior support of \(D_k\), away from black faces, has expansion at most \(k'+C_b\). The original solution limit is taken first for each fixed test; the reparametrization limit is a separate subsequent limit.

This property excludes contact with a white face: the reversed white face itself would be an exterior support, with black expansion \(-c_w>0\). The radial height bound makes \(D_k\) compact on the end, since \(k<0\). Apply Lemma 20 in the original connected truncation \(\Omega_R\), with the original boundary as required inner face and \(S_R\) as its only designated outer face. Its compact set is exactly \(D_k\cup\mathcal B_{c_b}\): neither the white regions nor omitted bounded components are adjoined. Every frontier point of this union outside \(\mathcal B_{c_b}\) lies in the reduced exterior away from white faces, and hence has the just-proved support bound. At any component of the smooth frontier of \(\mathcal B_{c_b}\), including components facing omitted regions, a support of the union also supports that black region, and comparison gives the bound \(c_b<k'+C_b\). Thus the union contains the required inner collar and meets all hypotheses of the lemma, whether or not it is connected. The lemma encloses it in a smooth trapped region at any threshold slightly larger than \(k'+C_b\) and still negative. It is therefore contained in the full black region at that larger threshold.

Right continuity at \(c_b=b_-+C_b\) lets us choose a threshold just above \(c_b\) whose full region still misses \(C_*\). Then choose \(k,k'\) close enough to \(b_-\) that the preceding enclosure uses a smaller threshold. A purported sequence in \(C_*\) with limiting height at most \(b_-\) places a limit point in \(D_k\cap C_*\), a contradiction. This proves uniform strict separation by the usual subsequence criterion.

For the other color apply the argument to \(-h,-K,-C\). Extend the original heights constantly by their low boundary values locally behind black faces before taking the reverse relaxed limit. The reversed black face is a forbidden support for the high sublevels. The ambient domain is now the fixed black complement in \(\Omega_R\), connected after the black pocket filling, with all black frontiers and \(S_R\) designated as outer faces. Adjoin only the full white region to the high sublevel. Supports on any full white frontier, including hidden components, are controlled by its smooth threshold equation. Visible black faces are excluded as just explained. A hidden black component has positive distance from the closure of the reduced domain: a fixed collar along its connected smooth face lies in a single adjacent complement component. Thus the union is disjoint from every designated outer face. Omitted components are not adjoined and introduce no other support frontier. Any pocket filling in the enclosure lemma is relative to these designated faces. Hausdorff right continuity at \(c_w\) gives the same contradiction. If a selected color is absent, right continuity means nearby larger-threshold regions are empty, and the enclosure contradiction is unchanged. ◻

Proof. In a black collar, test with \(h_0=b_-+\psi(s)\), where \(\psi(0)=0\) and \(\psi'>0\). At a comparison point the gradient is common to the solution and test, so \(t,l,d,\chi\) are computed using that gradient and the given value of \(Z\). The trace of the test is \[ P_s+aH_s+\chi K(\nu_s,\nu_s) +a\chi\left[ \frac{\mathop{\mathrm{Hess}}s(\nu_s,\nu_s)}{|\nabla s|} +|\nabla s|\frac{\psi''}{\psi'}\right], \tag{114}\] where \(H_s\) and \(P_s\) use \(\nu_s\). For slopes bounded above and below by positive constants, \[ d\le C(\tau_0/\ell)^2=C e^{-n\varepsilon},\qquad 1-a\le d,\qquad \chi\le d,\qquad n\chi=\frac{4nd}{4+pv}\ge2d. \tag{115}\]

Use Lemma 23 on the far leaf of a fixed short collar, and choose a small positive slope so that a lower test stays below the solution there. In this direction \(D_0\) vanishes. The stable-leaf inequality and \(C=C_b-k_Bs\) give a positive trace-minus-right-side residual at least \(k_Bs/2-O(d)\), apart from the last logarithmic-slope term in Equation (114). Choose \[(\log\psi')'=An\quad(0\le s\le1/n),\] nonnegative and smoothly cut off to zero by \(s=2/n\), with a fixed sufficiently large \(A\). By Equation (115) its contribution dominates \(O(d)\) on the initial interval. From \(s\ge1/n\) onward, the \(s\) margin dominates the exponentially small \(d\) error. The slope changes by at most the fixed factor \(e^{2A}\); choose its initial size after \(A\), keeping the whole test small. This constructs a lower barrier with strictly positive boundary derivative.

All constants in this choice are independent of \(n\) and \(R\). The far leaf is fixed with the domain; height separation gives one fixed \(\eta'\) there for every sufficiently large \(n\) and \(R\ge R_0(n)\). Choose \(A\) from the fixed geometric residual constants first, and only then choose the initial slope using \(\eta'\), \(k_B\), the collar width and the fixed ratio \(e^{2A}\). Thus these comparisons introduce no arbitrary fixed-\(n\) constants into the polynomial bounds \(\Pi_n\).

For an upper barrier choose a large positive slope and the opposite logarithmic derivative. The smooth leaf and offset variations are \(O(s)\), whereas the height increase dominates them; the negative logarithmic-slope term controls the initial \(d\) errors. The global height bound supplies comparison on the far leaf. Strict height monotonicity gives the comparisons throughout the collar. Equality at its initial face then gives the two black boundary derivative bounds in Equation (113). Reverse signs for white faces. These barriers near the appropriate faces, together with height separation off their own-color collars, give \(b_-\le h\le b_+\) on the required compact core.

In the third stage, start at the current assigned height \((1-\zeta)b\) and use lower and upper tests with large negative and positive slopes, respectively. Taper the magnitude of each slope using the negative logarithmic derivative just described. The limiting directional inequalities (104) give the respective residual signs at the face. At fixed boundary height the errors are \(O(s+d+\chi)\); in particular, the term \(\mathop{\mathrm{tr}}_{A_\chi}K\) is compared using the common leaf normal. The height shift controls the \(s\) error, and the logarithmic-slope term, using Equation (115), controls \(d+\chi\). This gives both boundary derivative bounds in the third stage. All uses of \(l\ge\ell\) occur at a common test gradient, so a ceiling for \(Z\) is unnecessary. ◻

Proof. Set \(W=w/\sqrt D\). By the two-sided normal-slope bounds (113), \(w\cdot\nu\) has a definite sign on each face and \(a\ge1/2\) there. The corresponding signed fluxes of \(uW=Lw\) and \(\sqrt D W=w\) are \(La\) and \(a\). Apply the divergence theorem to these fields, multiplied by the collar cutoff and \(\varphi\), reversing the sign on the white collars. The estimates needed for the resulting volume terms are \[\begin{gather*} |W(\varphi)|\le |\mathrm d\varphi|_{A_d} \le \Pi_n|\mathrm d\varphi|_{A_\chi}, \tag{119}\\ \frac{|\mathop{\mathrm{div}}(uW)|}{u} +\frac{|\mathop{\mathrm{div}}(\sqrt D W)|}{\sqrt D} \le\Pi_n(1+\sqrt{\mathcal T}). \tag{120}\end{gather*}\] For the second inequality, direct differentiation gives \[\mathop{\mathrm{div}}w=\mathop{\mathrm{tr}}_{A_d}H^f+d\,p\,w(t).\] In \(\mathop{\mathrm{div}}(Lw)/u\) every derivative is multiplied by \(\sqrt d\); in particular the longitudinal Hessian contribution is controlled by \(\sqrt d\,|q|\). Derivatives of \(L\) contribute \((p+2)\sqrt d\,w(t)\). These and the transverse terms are bounded by (92); replacing \(H^f\) by \(S_1-K\) costs a bounded compactly supported term. The same calculation applies to the second quotient in (120). Derivatives of the fixed collar cutoffs have bounded cost. This proves both trace estimates.

At assigned first-stage heights, the two colors both satisfy \(w_\nu(F-P_B)=aH\). Consequently the physical boundary expression is \[\mathcal B=(a-1)H-N_Bd =-d\left(N_B+\frac{H}{1+a}\right).\] The omitted normal drift in the first homotopy stage has size \(O(\chi)\le O(d)\), because \(\mathrm df\) is normal on a face and \(A_\chi\) has normal eigenvalue \(\chi\). The prefactor in the homotopy boundary condition lies in \([0,1]\). Thus its flux has size \(O(ud)\). Finally, (113) and \(l\ge\ell\) give \(D^{-1/2}\le C\tau_0/\ell\), proving (118). ◻

Proof. Only an upper bound for \(Z\) remains to be proved. Indeed \(Z=t+\tfrac18\log D\ge-\varepsilon\), and an upper bound for \(Z\), together with \(t\ge-\varepsilon\) and \(l\ge\ell\), bounds \(D\), \(t\) and \(\sigma\). The height estimate already bounds \(f=h/\tau_0\) at fixed \(n\).

Proof. We organize the proof so that neither a derivative nor a continuity modulus of \(\chi\) enters.

Proof. Use uniform patches on the compact core and on the end, including boundary normal patches. Boundary components carrying different conditions are separated; the patches are chosen to meet at most one face. At every stage the trace equation is \[ A_\chi:\mathop{\mathrm{Hess}}f =\frac{\sqrt D}{l}\bigl(F-\mathop{\mathrm{tr}}_{A_\chi}K\bigr). \tag{138}\] On the bounded variable range, \(l/\sqrt D\) is bounded below and \(\chi\) has a positive lower bound at fixed parameters. The right side is bounded, and the face values of \(f\) are componentwise constant. Lemma 27 consequently gives a uniform Hölder bound for \(D_xf\) and \[ \int_{B_r}|D_x^2f|^2\,\mathrm dx\leq Cr^{1+2\alpha}. \tag{139}\]

Proof. We first work with smooth input \(Z\) and obtain estimates depending only on its \(C^{1,b_{\rm reg}}\) norm. Approximation will then give the asserted solve for general inputs in that space. Write the normalized trace equation as \[ A_\chi:\nabla^2f=G_f, \qquad G_f=\frac{\sqrt D}{l} (F-\mathop{\mathrm{tr}}_{A_\chi}K). \tag{147}\] Here \(F\) includes the modification appropriate to the current stage. For an auxiliary parameter \(s\in[0,1]\), interpolate between this equation and \(\Delta f=\tau_0f\): \[ A^{(s)}:\nabla^2f=G^{(s)},\qquad A^{(s)}=(1-s)I+sA_\chi,\qquad G^{(s)}=(1-s)\tau_0f+sG_f. \tag{148}\] Keep the current boundary values throughout this interpolation. At zero gradient \(A_\chi=I\) and \(\mathop{\mathrm{tr}}_{A_\chi}K=\mathop{\mathrm{tr}}K\). The strict monotonicity of \(F\) in \(h\) therefore gives a uniform height bound by the maximum principle, for every \(s\) and every bounded input set.

We next establish a global gradient bound for this scalar interpolation. If \(Z\) is bounded and \(\sigma\to\infty\), then \(t\to-\infty\). In that range \(p=n\) and \(l=e^{nt}\), and the defining relation between \(t\) and \(Z\) is exactly \[ e^{8Z}=e^{8t}+e^{(2n+8)t}\sigma^2. \tag{149}\] It follows, uniformly for bounded \(Z\), that \[d\asymp\sigma^{-8/(n+4)},\qquad \chi\asymp\sigma^{-8/(n+4)},\qquad \frac{\sqrt D}{l}\asymp\sigma.\] Because \(n>4\), the exponent \(8/(n+4)\) is less than one. On the entire bounded-height range we consequently have \[ \chi\ge\frac{c}{1+\sigma},\qquad |G_f|\le C(1+\sigma). \tag{150}\] The same inequalities hold for the longitudinal eigenvalue \(\chi^{(s)}=1-s+s\chi\) and the right side \(G^{(s)}\), with uniform constants. The two transverse eigenvalues remain exactly one.

Let \(r_0\) be distance from a boundary component in a fixed smooth collar, and set \[ \psi(r_0)=\frac1k\log(1+B_0r_0), \qquad \psi''=-k(\psi')^2. \tag{151}\] Use \(f_{\mathrm{face}}+\psi\) as an upper barrier and \(f_{\mathrm{face}}-\psi\) as a lower barrier. At a possible contact their gradient has length \(q'=\psi'\), and the longitudinal second derivative has magnitude \(k(q')^2\). By (150), its signed contribution dominates all forcing and collar-curvature terms: \[\chi^{(s)}\psi''+O(q') \le-\frac{kc(q')^2}{1+q'}+Cq'.\] Choose \(k\) large relative to the structural constants, then a collar width \(\delta'\) small enough that \(1/(2k\delta')\) exceeds the required slope threshold. Finally choose \(B_0\delta'\ge1\) so large that \(\psi(\delta')\) exceeds the height oscillation. Throughout the collar \(\psi'\ge1/(2k\delta')\), so the differential inequalities have strict sign; height monotonicity rules out a positive comparison maximum. This proves \[ |f-f_{\mathrm{face}}|\le\psi(r_0) \quad(0\le r_0\le\delta'). \tag{152}\] The assigned values are constant on each face, and \(\delta'\) can be chosen below the separation of distinct boundary components.

For the interior estimate maximize \[f(x)-f(y)-\psi(\mathop{\mathrm{dist}}(x,y))\] over pairs at distance at most \(\delta'\). The distance may be computed in a fixed smooth extension of the metric; choose \(\delta'\) below its uniform injectivity radius. Pairs with one point on the boundary are controlled by (152), since the distance to that boundary component is no greater than the pair distance. Pairs at distance \(\delta'\) are controlled by \(\psi(\delta')\ge\operatorname{osc}f\). A positive maximum would therefore occur at an interior pair with \(r_0=\mathop{\mathrm{dist}}(x,y)>0\). The two gradients there are parallel transports of the common vector of length \(q'=\psi'(r_0)\). Simultaneous variations in parallel transverse directions, using parallel trial fields in the second variation of distance, give \[\mathop{\mathrm{tr}}_\perp\nabla^2f(x)-\mathop{\mathrm{tr}}_\perp\nabla^2f(y) \le Cr_0q'.\] Separate longitudinal variations give \[\nabla^2f(x)(e,e)\le\psi'',\qquad \nabla^2f(y)(e,e)\ge-\psi''.\] Subtracting (148) at the two points yields \[\begin{align*} -2C(1+q') &\le G^{(s)}(x)-G^{(s)}(y)\\ &\le Cr_0q'+(\chi_x^{(s)}+\chi_y^{(s)})\psi''\\ &\le Cr_0q'-\frac{2kc(q')^2}{1+q'}. \end{align*}\] Our choices of \(k\) and the lower threshold for \(q'\) make this impossible. Notice that the two longitudinal eigenvalues need not coincide or have a known modulus of continuity. Thus \(|f(x)-f(y)|\le\psi(\mathop{\mathrm{dist}}(x,y))\) for nearby pairs, proving a global Lipschitz bound on each bounded input set.

The interpolation is now uniformly elliptic. Lemma 27 applies to its axial matrix and bounded forcing, first giving a positive Hölder modulus for \(\nabla f\). Schauder estimates give \(C^{2,\theta}\) and then, by differentiating after this first gain, \(C^{2,b_{\rm reg}}\) and \(C^{3,b_{\rm reg}}\) bounds. More explicitly, the first differentiated estimate gives \(C^{3,\min(\theta,b_{\rm reg})}\); hence \(\nabla f\) is Lipschitz, allowing the original coefficients to be estimated in \(C^{b_{\rm reg}}\), followed by the stated full exponent. All these bounds are uniform in \(s\).

For fixed input \(Z\), the principal matrix is independent of the height \(f\), while \[\partial_fG^{(s)} =(1-s)\tau_0+s\frac{\sqrt D}{l}\tau_0\partial_hF>0.\] The linearized Dirichlet operator thus has strictly negative zeroth-order coefficient when written with everything on the left. The maximum principle and linear elliptic Dirichlet theory make it invertible (Gilbarg and Trudinger 2001). Starting at \(\Delta f=\tau_0f\), openness follows from the implicit function theorem, and the bounds above give closedness. This proves existence for \(s=1\). At a positive maximum of the difference of two solutions, their gradients and principal matrices agree, while strict height monotonicity contradicts the Hessian inequality; this proves uniqueness.

For a general \(C^{1,b_{\rm reg}}\) input, choose smooth invariant inputs \(Z_j\) with the same zero outer data, uniformly bounded in \(C^{1,b_{\rm reg}}\), and converging in \(C^{1,b'}\) for every \(b'<b_{\rm reg}\). Such an approximation follows by extension and smoothing in boundary charts, correction of the outer trace, and averaging under the circle action. Convergence in the full \(C^{1,b_{\rm reg}}\) norm is neither asserted nor needed. The smooth scalar solutions have the uniform \(C^{3,b_{\rm reg}}\) bounds just proved. Compact embedding gives a subsequence converging in \(C^{2,b_{\rm reg}}\) to a solution for \(Z\); the third-derivative Hölder seminorm bound passes to the limit by lower-exponent compactness. Uniqueness identifies this limit and removes dependence on the approximation. Finally, for any convergent sequence of actual inputs in \(C^{1,b_{\rm reg}}\), the same compactness and uniqueness argument gives continuous \(C^{2,b_{\rm reg}}\) dependence, also at the matching endpoints of the four homotopy stages. Symmetry follows from uniqueness. ◻

Proof. Use the Banach space \(X_R\) of invariant \(C^{1,b_{\rm reg}}(\overline\Omega_R)\) functions with outer value zero. For each input \(Z\), solve the trace equation by Lemma 29. Compute \(t,u,A_\chi,\Xi\) and all boundary expressions from this input and its solved \(f\). Freeze them, replacing only the differentiated \(Z\) in the principal flux by the output unknown \(\widetilde Z\). In the first three stages, for example, the linear problem is \[\mathop{\mathrm{div}}\bigl(4uA_\chi\nabla\widetilde Z +uA_\chi\lambda K(w,\cdot)\bigr)=u\Xi,\] with the prescribed frozen inner flux and zero outer Dirichlet value. The fourth stage uses its stated simultaneous source and boundary scaling.

On bounded input sets the coefficients are uniformly elliptic. The nonempty outer Dirichlet face makes this mixed problem coercive; there is no flux compatibility condition. Its coefficients, drift, and inner flux data are at least \(C^{1,b_{\rm reg}}\), and its source is at least \(C^{0,b_{\rm reg}}\). Linear boundary regularity therefore bounds the output in \(C^{2,b_{\rm reg}}\) (Agmon et al. 1959; Gilbarg and Trudinger 2001). Together with continuous dependence of the scalar solve, this defines a continuous compact map \(T_s:X_R\to X_R\), continuous in the full homotopy parameter \(s\). The derivative count is worth specifying: \(f\in C^{3,b_{\rm reg}}\) and \(Z\in C^{1,b_{\rm reg}}\) imply that \(t\), \(uA_\chi\) and the drift are \(C^{1,b_{\rm reg}}\), while the quadratic expression \(\mathcal T\) is \(C^{0,b_{\rm reg}}\). Uniqueness of both solves preserves invariance. At a fixed point, \(Z\in C^{2,b_{\rm reg}}\) and \(f\in C^{3,b_{\rm reg}}\) initially. Then the frozen coefficients improve to \(C^{2,b_{\rm reg}}\) and the source to \(C^{1,b_{\rm reg}}\), giving \(Z\in C^{3,b_{\rm reg}}\); the trace solve next gives \(f\in C^{4,b_{\rm reg}}\). Repeating this triangular bootstrap makes each fixed point smooth. Proposition 28 supplies the uniform classical bounds for these smooth fixed points.

Let \(t_s[Z]\) denote the value of \(t\) after the scalar solve and define the relatively open subset of \([0,1]\times X_R\) \[\mathcal O=\{(s,Z):t_s[Z]>-\varepsilon \text{ on }\overline\Omega_R,\quad\|Z\|_{X_R}<M\}.\] Choose \(M\) larger than the fixed-point bound supplied by Proposition 26 and Proposition 28. Lemma 22 excludes fixed points on the floor boundary; the classical estimates exclude them on the norm boundary. Fixed points in the closure of \(\mathcal O\) form a compact set, because the homotopy maps bounded sets to relatively compact sets. Leray–Schauder degree is therefore invariant on the moving sections \(\mathcal O_s\) (Leray and Schauder 1934). One way to see the minor moving-domain point is to cover this compact fixed-point set by finitely many product neighborhoods lying in \(\mathcal O\), subdivide the parameter interval accordingly, and apply homotopy invariance and excision on those fixed neighborhoods. No control of \(T_s\) on unbounded input sets is needed.

At the terminal endpoint the trace comparison gives \(f=0\), and the linear second solve has identically zero output. The zero input is interior to \(\mathcal O_1\), so its degree is one. The physical endpoint consequently has a fixed point with \(t>-\varepsilon\). For this fixed \(n\), let \(R\to\infty\) through the admissible truncations. The estimates of Propositions 26 and 28 are uniform on every compact subset, including inner-boundary charts. A diagonal subsequence converges smoothly there, giving the asserted physical solution and the limiting bound \(t\ge-\varepsilon\). ◻

Proof. Beyond the support of \(K,C,D_0,D_1\), the trace equation is \[A_\chi:\nabla^2 f=\tau_0\frac{\sqrt D}{l}f.\] On the bounded variable range at fixed \(n\) its principal matrix is uniformly elliptic and its height coefficient has a positive lower bound. Slowly decaying exponentials \(Ce^{-c(r-r_*)}\) are positive supersolutions for the operator obtained by moving the height term to the left, if \(c>0\) is small. Choose \(C\) to dominate the data on \(r=r_*\). The zero outer data allow the same barriers on every truncation; applying them to both signs and then exhausting gives exponential decay of \(f\). For the derivative assertion, freeze the coefficients in this homogeneous linear equation along the solution. Proposition 28 bounds their \(C^m\) norms on uniform unit patches, independently of \(R\), for every fixed finite \(m\). Local homogeneous estimates therefore give \(\|f\|_{C^{j,\alpha}(B_1)}\le C(n,j)\|f\|_{C^0(B_2)}\); on outer half-patches the same estimate uses the zero Dirichlet condition. Each finite \(j\) requires only finitely many of the already bounded coefficient derivatives. Thus every required derivative of \(f\) decays exponentially; no decay of the derivatives of \(Z\) is needed at this step.

For clarity, put \(E=t-Z=-\tfrac18\log D\). Then \(E\), \(D-1\), \(v\), \(A_\chi-I\), and their required derivatives are \(O_n(e^{-2cr})\), while \(w,H^f,F=O_n(e^{-cr})\) with derivatives. The tensor expression for \(\mathcal T\) consequently gives \[\mathcal T=4[p(Z)+1]|\mathrm dZ|^2+O_n(e^{-2cr}).\] Every remaining term contains two factors from \(H^f,F,w\), or a graph error multiplying bounded derivatives of \(Z\). In particular, \(u-L(Z)\) and \(p(t)-p(Z)\) have the same exponential bound, and \[V=4L(Z)\nabla Z+E_V,\qquad \mathop{\mathrm{div}}E_V=O_n(e^{-2cr}).\] Here the divergence uses only the radius-uniform unit-patch bound on \(\nabla^2Z\). The height term \(\tau_0a\sigma\) and the penalty term \(v\rho_1\) are exponentially small as well. There are no persistent \(K\) terms, since \(K\) vanishes on this end; derivatives of the profiles have bounded fixed-\(n\) coefficients and multiply the stated graph errors. Consequently the end divergence equation takes the form \[ \Delta Z+ \bigl[p(Z)+2-\delta_0(p(Z)+1)\bigr]|\nabla Z|^2 =O(r^{-4}). \tag{154}\] This also holds on the truncations, with uniform constants. More precisely, the right side is \[\frac14\rho-\frac14C_nm_0(Z)\rho_0+O_n(e^{-2cr}),\] which is \(O(r^{-4})\) without any prior decay estimate for \(Z\).

Put \(b(z)=p(z)+2-\delta_0(p(z)+1)\) and choose \(\Phi(0)=0\), \(\Phi'(z)=\exp(\int_0^z b(q)\,\mathrm dq)\). On the bounded solution range, \(\Phi'\) and its reciprocal are bounded. Then \(Y=\Phi(Z)\) satisfies \(\Delta Y=O(r^{-4})\). For \(0<\eta<1\), the positive function \(r^{-1}-r^{-1-\eta}\) has Laplacian \(-\eta(1+\eta)r^{-3-\eta}+O(r^{-4})\) in the AF metric. Multiples of this function dominate both the source and the bounded inner-end data, and are positive at the zero outer Dirichlet face. Comparison on the truncations proves \(Y=O(r^{-1})\), hence \(Z=O(r^{-1})\) after exhaustion.

The penalty therefore vanishes sufficiently far out. In the transformed equation the remaining nonexponential terms have smooth weight coefficients of order \(-4\) and bounded smooth factors in \(Z\). On rescaled annuli, a \(W^{2,p}\) estimate with \(p>3\) first bounds the rescaled gradient and its Hölder modulus. Schauder estimates then bound the rescaled second derivatives. The exponentially small graph errors are controlled in these estimates by the unit-patch bounds of every required finite order. This yields \(Z,t=O_2(r^{-1})\).

It follows that \(\mathop{\mathrm{div}}V=u\Xi\) is integrable and \(V=4\nabla t+O(r^{-3})\). The divergence theorem on \(\Omega_R\) gives \[\int_{S_R}V_\nu=\int_{\Omega_R}u\Xi+\int_BV_\nu,\] because the outward normal of \(\Omega_R\) on \(B\) is \(-\nu\). This proves existence of the flux limit. The graph term has exponentially decaying ADM contribution. For the conformal factor \(e^{4t}\) the ADM change is \(-(2\pi)^{-1}\lim\int_{S_R}\partial_\nu t\), which is precisely \(-\mathfrak F_n/(8\pi)\). This proves (153). ◻

Proof. At the physical endpoint, (118) bounds the negative inner flux by \(C(\tau_0/\ell)\int_B L\). Apply (116) with a cutoff equal to one near \(B\). Since \(\rho\) is bounded below on the collars, \[1+\sqrt{\mathcal T}\le C(\rho+\delta_0\mathcal T)\] there. As \(\Pi_n\tau_0/\ell\to0\), this cost absorbs into, for example, half of the positive bulk integral of \(u(\rho+\delta_0\mathcal T)\). These are polynomial estimates; none of the arbitrary fixed-\(n\) regularity constants enters this absorption.

On the penalty support, \(-\varepsilon\le t\le-\varepsilon+1/n\), so \(l\) and \(L\) are comparable to \(\ell\). Directly from the definitions, \[ u\le C(\ell+\ell^2\sigma),\qquad uv\le C\ell^2\sigma,\qquad ua\sigma=Ll\sigma^2\ge c\ell^2\sigma^2. \tag{155}\] Therefore \[um_0\mathcal P\le \Pi_n\bigl[\ell\rho_0+ \ell^2\sigma(\rho_0+\rho_1)\bigr].\] Young’s inequality against the remaining positive \(\delta_0\tau_0ua\sigma\) term yields \[ um_0\mathcal P\le \frac{\delta_0\tau_0}{2}ua\sigma+ \Pi_n\left[\ell\rho_0+ \frac{\ell^2}{\tau_0}(\rho_0^2+\rho_1^2)\right]. \tag{156}\] The displayed remainder has vanishing total integral. Indeed \(\rho_0\asymp r^{-4}\) is integrable, \(\rho_0^2\) is integrable, and \(\rho_1^2\asymp r^{-4\gamma}\) is integrable in dimension three because \(\gamma>3/4\). Finally \[\ell=e^{-n\varepsilon},\qquad \frac{\tau_0}{\ell}=\ell^{1/2},\qquad \frac{\ell^2}{\tau_0}=\ell^{1/2},\] so both small factors dominate every polynomial loss in \(n\). Insert these bounds in (153) to obtain the proposition. ◻

Completion of the proof of Theorem 18. At the physical endpoint \(F=\tau_0f+C\), whence \(w(F)=\tau_0a\sigma+w(C)\). Substituting (101) into (89) shows that the remainder after \(8\pi(\mu+J_0(w))\) is \[(1-\delta_0)\mathcal T+ (1-\delta_0)\tau_0a\sigma+m_0\mathcal P +w(C)-F\mathop{\mathrm{tr}}K-\rho.\] The first three terms are nonnegative, and the height and offset bounds give \[w(C)-F\mathop{\mathrm{tr}}K-\rho \ge-|\nabla C|-|F\mathop{\mathrm{tr}}K|-\rho\ge-\mathfrak m/2.\] This is the scalar-curvature conclusion of the theorem. Combining the physical boundary condition with (94) gives \(H+4\partial_\nu t=-N_B\), so \[H_{\hat g}=-e^{-2t}\sqrt d\,N_B<0.\] The metric inequality \(\hat g\ge e^{-4\varepsilon}g\) guarantees completeness with the inner boundary included. Lemma 31 supplies the required end regularity and finite energy, and Proposition 32 supplies its upper bound. All choices respect axial invariance. The limits are taken in the order established above: fix the prepared data, selected domains, cutoff shapes and \(\varepsilon\); choose large \(n\) and form its profiles; solve and exhaust \(R\) at that fixed \(n\); then let \(n\to\infty\) in the scalar energy estimate. No convergence of the deformation metrics as \(n\to\infty\) is needed. ◻

Proof. Extend the geometry smoothly and equivariantly across \(S\) into an auxiliary cap, and require each bounded inner set to contain that cap up to \(S\). Count its full frontier, including coincidence with the obstacle. First impose an outer sphere obstacle. Perimeter compactness gives a minimizer. It has a \(C^{1,1}\) frontier, smooth and minimal off the inner obstacle, by smooth-obstacle perimeter regularity in dimension three (Huisken and Ilmanen 2001, Regularity Theorem 1.3(iii)).

The outer sphere can be removed uniformly for metrics in the paths under consideration. Large coordinate spheres are strictly mean convex, and the competitor \(S\) gives a uniform area upper bound. If a free minimizing frontier had a point at radius \(r_j\to\infty\), its portion in an asymptotic ball of radius \(\varepsilon r_j\), for a fixed small \(\varepsilon>0\), would have area at least \(c r_j^2\) by the interior density estimate for perimeter minimizers. This contradicts the area upper bound. Thus every minimizer stays in one compact set. Filling bounded pockets decreases perimeter and preserves admissibility.

Among minimizers in this compact set choose one with largest filled volume. The perimeter inequality for union and intersection shows that both the union and the intersection of two minimizing inner sets are again minimizers. Rotating a maximum-volume minimizer and taking its union with the original one therefore leaves it unchanged modulo null sets. It is invariant. Smooth outward approximation of its regular graphs, or equivariant smoothing of an outward defining function, gives smooth invariant enclosing cuts with convergent area. Invariant areas are preserved by polarization. The original minimum-area hypothesis consequently gives \(a(g)\geq A\), while \(S\) gives the reverse inequality.

For a sequence \(d_j\to0\), choose invariant minimizers \(T_j\) for \(\widetilde g_{d_j}\). Compact containment and perimeter compactness give a subsequence converging to a filled inner set \(T\). Uniform equivalence of the metrics, lower semicontinuity, and comparison with a background minimizer show that \(T\in\mathcal T_0\) and that the perimeters converge. Strict convergence of the vector measures representing the perimeter normals then implies convergence of their area and tangent-plane measures. In particular, the integral defining \(A'_T\) is continuous under this convergence.

On the common compact set the expansion of area is uniform: \[\mathop{\mathrm{Area}}_{\widetilde g_d}(\partial T) =\mathop{\mathrm{Area}}_g(\partial T)+dA'_T+O(d^2)\mathop{\mathrm{Area}}_g(\partial T).\] Comparing the perturbed minimum first with a minimizing member of \(\mathcal T_0\), and then with its own minimizer \(T_d\), proves the upper and lower one-sided bounds in Equation (161). The same compactness argument at \(d=0\) proves compactness of \(\mathcal T_0\). ◻

Proof. Let \(\mathcal B^+\) be the one-point compactification of \(\mathcal B\). For a variation \(v\) consider the continuous test triple \[L(v)=\left( \frac{C'[v]}{w_0},\ -\theta_+'[v],\ \left\{cA'_T[v]-16\pi E'_g[v]\right\}_{T\in\mathcal T_0} \right)\] in \[\mathcal X=C(\mathcal B^+)\times C(S)\times C(\mathcal T_0).\] Continuity at the added point follows from Equation (165) and \(\delta<1\); continuity of the last block follows from Lemma 34.

The linear space \(L(V)\) misses the open cone of triples strictly positive everywhere. Indeed a variation in this cone has a positive uniform margin in all three blocks. It has a smooth path realizing its derivative with \[C_d=dC'+O(d^2w_0),\qquad \theta_+(d)=d\theta_+'+O(d^2),\] uniformly on the indicated domains. For completeness, realize a finite sum of compact directions, the prototype, and the strict direction by first making the linear compact and prototype changes, and then applying the displayed exponential scalings with parameter \(a_{\rm dir}d\). On a compact set the Taylor remainders are uniform, also through the axis by the polar orders. On the end the new metric tails are \(O_2(r^{-1})\); the second-order scalar-curvature remainder is \(O(d^2r^{-4})\), as are or dominate all other constraint remainders. Since \(r^{-4}=O(w_0)\), this proves the stated estimate. The end curvature remains \(O(r^{-3-\delta})\), the constraints remain integrable, and the ADM energy is differentiable. All other assumptions of the enlarged class in Corollary 19 persist. For small \(d>0\) the path is therefore feasible. Equation (161) and the strictly positive objective block make the derivative of \[16\pi\left[ E_{g_d}-F_*\left(\sqrt{\frac{a(g_d)}{4\pi}}\right)\right]\] strictly negative at its zero background value, contradicting Equation (162).

Separation of an open convex cone from a linear subspace gives a nonzero continuous positive functional on \(\mathcal X\) annihilating \(L(V)\). This separation requires no closed-range assertion for \(L\). Riesz representation gives three nonnegative finite measures. The measure of the objective block has positive total mass: otherwise evaluation on the strict direction, whose first two blocks are strictly positive, forces both remaining measures to vanish. Divide by that positive mass to obtain \(\omega\). If \(\overline\Lambda\) denotes the resulting first measure, set \(\mathrm d\Lambda=w_0^{-1}\mathrm d\overline\Lambda\) on \(\mathcal B\). It is locally finite. Its atom at the added point gives \(z_0a_{\rm dir}\), since the first test block has value \(8a_{\rm dir}\) there. This proves Equation (166). Averaging the identity over the compact symmetry group gives the last assertion. ◻

Proof. Let \(\sigma_0\) be an arbitrary smooth invariant normal speed supported strictly inside \(\Omega\). Take a short local Einstein–Maxwell Cauchy development of the original data about its compact support. Local hyperbolic existence and uniqueness propagate the symmetry. A compact change of its spacelike slice, followed by polarization, gives exact constraint variations \[ C'=0,\qquad \dot g=2\sigma_0k. \tag{167}\] To justify the second identity, differentiate the orthogonal horizontal projection defining \(h\). Terms from differentiating the horizontal lifts pair to zero by orthogonality. Differentiating the fiber length gives the remaining fiber component, so the polarized variation is precisely \(2\sigma_0k\).

The potentials in these tests can be chosen with compact variation and unchanged axis constants. Maxwell’s equations give the closed spacetime orbit forms defining \(\psi,\chi\), which are integrated normally in the short slab. With these potentials fixed, the orbit form defining \(z\) is closed as well: its exterior derivative vanishes on every spacelike meridional plane by the axial momentum equation on each tilted slice. Such planes test every two-form locally in the three-dimensional orbit space. Normal integration therefore extends \(z\) with its initial constants. These constructions preserve the polar orders. Only the interior neighborhood of the chosen support is used.

All end and boundary variations, and \(a_{\rm dir}\), vanish in these tests. Equations (166) and (167) give \[ \int_{\mathcal T_0}\int_{\partial T} \sigma_0\,\mathop{\mathrm{tr}}_{\partial T}k\,\mathrm dA_g\,\mathrm d\omega(T)=0. \tag{168}\] The common-plane property above makes the tangential trace a single measurable function on the union of free frontiers. The aggregate area measure is positive. Equation (168), first for invariant tests and then for all tests by averaging, says that this function vanishes almost everywhere for the aggregate measure. Integrating its absolute value and using Fubini shows that, for \(\omega\)-almost every \(T\), the trace vanishes on almost every point of its free frontier. Continuity makes it vanish everywhere on each smooth free portion. Such portions are minimal and hence satisfy the MOTS equation.

The same equation holds almost everywhere across contact with \(S\). In tangent graphs the difference between the frontier and the obstacle, and its first derivatives, vanish on the contact set. Since the graphs are \(C^{1,1}\), differentiability and the density theorem imply that their second derivatives also agree at almost every contact point. Their outward orientations agree by obstacle inclusion. The frontier therefore solves the smooth uniformly elliptic MOTS graph equation almost everywhere across contact. In a graph chart, write this as \[a^{ij}(x,f,\mathrm df)\partial_i\partial_jf =b(x,f,\mathrm df).\] The \(C^{1,1}\) graph is a strong \(W^{2,\infty}\) solution; bounded slope makes the principal matrix uniformly elliptic. Evaluated on \((x,f,\mathrm df)\), both the coefficients and the right side are Lipschitz. Interior Schauder regularity for this linear equation therefore gives \(f\in C^{2,\alpha}\) for every \(\alpha<1\) on a smaller chart. Differentiation and elliptic bootstrap then make the graph smooth. Strong comparison implies local coincidence whenever it touches \(S\). Connectedness then forces a touching component to be all of \(S\); its total area is \(A\), so it has no further component.

If the frontier is disjoint from \(S\), it is a smooth interior enclosing MOTS. For an invariant surface the original and polarized expansions agree. The filled-frontier convention removes any spurious pocket component, so this is an enclosing surface forbidden by the exact outermostness hypothesis of Theorem 2. Thus almost every minimizing cut in the multiplier is \(S\). ◻

Proof. Take the divergence of Equation (169) together with Equation (170). The scalar principal symbol is \(|\xi|^2\); the vector symbol is \[\frac12\bigl(|\xi|^2I+\xi\otimes\xi\bigr).\] All couplings are lower order. This is an elliptic system with smooth coefficients on the rotational three-space. Distributional elliptic regularity therefore proves interior smoothness, including at the axis.

We record the additional metric equation to justify continuation and the end estimates. For a pure metric variation the second-order term is \(\mathop{\mathrm{Hess}}N-(\Delta N)g\). The gravitational lower-order terms are smooth linear expressions bounded in norm by \[ C\left[(|\operatorname{Rm}|+|k|^2)|N| +|\nabla k|\,|Y|+|k|\,|\nabla Y|\right]. \tag{171}\] This follows directly from \[R'=\mathop{\mathrm{div}}\mathop{\mathrm{div}}\dot g-\Delta\mathop{\mathrm{tr}}_g\dot g-\langle\mathop{\mathrm{Ric}},\dot g\rangle,\] the algebraic variation of \((\mathop{\mathrm{tr}}k)^2-|k|^2\), and the \(\nabla k*\dot g+k*\nabla\dot g\) terms in the momentum variation. Terms due to varying the volume disappear because the background constraint residual is zero.

The matter terms, while initially written off the axis, have smooth tensor coefficients. Up to constant factors and smooth trace terms, they contract against \[ N\left(D_i^\sharp\otimes D_i^\sharp+ h_i|D_i|^2e\otimes e\right), \qquad s_iD_i(Y)e\otimes e. \tag{172}\] Free compact variations of \(s_i\) give \[ D_i(Y)=-Ns_i. \tag{173}\] Thus the second tensors become \(-Ns_i^2e\otimes e\), which are smooth by the polar orders. For \(i=1,2\), the radial and azimuthal coefficients of the first tensor have the same leading value on the axis and smooth even parity; their difference is \(O(y^2)\). The radial-longitudinal coefficient is smooth odd. In Cartesian polar disks these are exactly the conditions for a smooth invariant tensor. For \(i=3\), the higher vanishing orders directly give the same conclusion. This proves smoothness of the full metric equation through the axis; the matter terms are bounded by \(C|N|\sum_i(|D_i|^2+s_i^2)\).

Taking the trace of the metric equation and substituting back determines all components of \(\mathop{\mathrm{Hess}}N\) from the first jets of \(N,Y\). Differentiating Equation (169) and commuting covariant derivatives likewise expresses \(\nabla^2Y\) in those first jets, with coefficients formed from \(k,\nabla k,\operatorname{Rm}\). Hence \[(N,Y,\nabla N,\nabla Y)\] satisfies the asserted homogeneous first-order system. In a collar of \(S\), integrate its normal ordinary differential equation from an interior collar slice. Smooth dependence on the tangential parameters gives a smooth one-sided extension to the boundary.

There might initially remain a separate boundary-supported part of the moment measures. Take \(\dot k=0\) and a compact metric variation whose value and first jet vanish on \(S\), with arbitrary second normal derivative of its tangential trace there. Neither the area nor the expansion derivative sees this second jet. After integration of the smooth interior adjoint, its boundary terms also vanish. In the residual at \(S\), only the scalar-curvature variation tests the prescribed second jet. Equation (166) therefore forces the boundary part of \(N\) to vanish. The measure inequality \(|Y|\leq N\) then removes the boundary part of \(Y\). ◻

Proof. Set \(F=(N,Y)\). The first-jet equations and the original end bounds give \[ |\partial^2F| \leq C r^{-2-q'}|F|+C r^{-1-q'}|\partial F|. \tag{174}\] Here the matter coefficients have order at least \(r^{-4}\), so any \(q'\in(1/2,1)\) is allowed. Radial integration with Gronwall’s inequality, applied to \(|F|/r+|\partial F|\), first gives \(F=O(r)\) and \(\partial F=O(1)\). The Hessian bound then gives a limiting gradient along each ray. Integrating the Hessian along paths of length \(O(r)\) on coordinate spheres shows that these limits agree. A nonzero linear term in \(N\) would contradict \(N\geq0\) in opposite asymptotic directions. Once that term vanishes, \(|Y|\leq N\) eliminates every linear term of \(Y\) as well. Consequently \[F=O(r^{1-q'}),\qquad \partial F=O(r^{-q'}).\] Reinsertion into Equation (174) gives \(\partial^2F=O(r^{-1-2q'})\). The gradient limits are zero, so \(\partial F=O(r^{-2q'})\). Since \(2q'>1\), radial integration gives finite value limits, and sphere paths show that there is a single such limit. One final insertion gives \[\partial^2F=O(r^{-2-q'}),\qquad \partial F=O(r^{-1-q'}),\qquad F-F_\infty=O(r^{-q'}).\]

Use the energy prototype in Equation (166). It vanishes near \(S\), and \(a_{\rm dir}=0\). Integrate the smooth adjoint over a large truncation. Scalar linearization gives the limiting constraint flux \(16\pi N_\infty E'_g\). Terms containing \(\nabla N\), the nonconstant part of \(N\), and momentum metric-variation boundary terms tend to zero by the preceding decay. The residual tail is integrable. Since \(E'_g=1/2\neq0\), the identity gives \(N_\infty=1\).

To eliminate the constant spatial translation, write \(g=\delta+H_0\) and put \(a=Y_\infty\) for this paragraph only. Equation (26) and scalar linearization give \[LH_0:=\partial_i\partial_jH_{0,ij} -\Delta_\delta\mathop{\mathrm{tr}}_\delta H_0=O(r^{-4}).\] For an affine function \(V\) define the Euclidean flux covector \[U(V;h')_i= V(\partial_jh'_{ij}-\partial_ih'_{jj}) -h'_{ij}\partial_jV+(\mathop{\mathrm{tr}}_\delta h')\partial_iV.\] It satisfies \(\partial_iU(V;h')_i=VLh'\). The shift equation gives \(\mathcal L_Yg=O_1(r^{-3})\). Subtracting \(\partial_aH_0\) and the pure Euclidean gauge \(\mathcal L_Y\delta\) leaves errors \(O_1(r^{-2-q'})\). The affine flux of these errors tends to zero. The affine flux of \(\mathcal L_Y\delta\) is identically zero after a smooth extension to the interior, because its scalar linearization vanishes. Thus the flux of \(U(V;\partial_aH_0)\) tends to zero.

On the other hand, differentiation of the definition of \(U\) gives \[U(V;\partial_aH_0)=\partial_aU(V;H_0)-U(\partial_aV;H_0).\] Euclidean integration by parts, with arbitrary smooth interior extensions, consequently yields the exact sphere identity \[ \begin{split} \int_{S_r}U(V;\partial_aH_0)\cdot n_\delta\,\mathrm dA_\delta &=\int_{S_r}(a\cdot n_\delta)V\,LH_0\,\mathrm dA_\delta\\ &\quad-(\partial_aV) \int_{S_r}U(1;H_0)\cdot n_\delta\,\mathrm dA_\delta. \end{split} \tag{175}\] The first term tends to zero; the second tends to \(-16\pi m\,\partial_aV\). Since \(m>0\), testing all affine \(V\) gives \(a=0\). This calculation uses just the hypotheses stated in the last part of the lemma.

Finally the prolonged shift equation, now with zero limits, gives \[|\partial^2Y| \leq C\bigl(r^{-2}|\partial Y|+r^{-3}|Y|+r^{-4}\bigr).\] Starting with \(Y=O_2(r^{-q'})\), integration from infinity first improves this to \(Y=O_2(r^{-1-q'})\), and then to \(Y=O_2(r^{-2})\). ◻

Proof. Choose a compact smooth invariant meridional vector field \(W\) with \(W|_S=\xi_0\nu\). Extend the data smoothly across \(S\) and take their Lie derivatives along \(W\). These are allowed compact first variations in Equation (166); they need not form a feasible path. Covariance gives \(C'=\mathcal L_WC=0\) on the exterior. This also holds at \(S\), since the exact residual vanishes on the interior and all its one-sided jets vanish at the boundary. No constraint condition on the extension is required. Naturality identifies the fixed-surface expansion of the pulled-back data with the expansion of the surface moved by \(W\) in the original data. Consequently the first area variation is \(\int_S H\xi_0\,\mathrm dA_g\), and the expansion variation is \(L_S\xi_0\), where \(L_S\) is the MOTS normal stability operator with principal part \(-\Delta_S\). Equation (166) and \(\mathrm d\lambda_S=2N\,\mathrm dA_g\) therefore give \[ 2L_S^*N=cH. \tag{178}\] Averaging allows arbitrary scalar tests. Outer area minimization implies \(H\geq0\) by outward first variation; the polarized and original mean curvatures of \(S\) agree. Thus \(L_S^*N\geq0\) with \(N\geq0\). To check the sign in the strong minimum principle, write locally \(L_S^*=-a^{ij}\partial_i\partial_j+b^i\partial_i+c_0(x)\). Choose a constant \(C_0\geq\max(0,\sup_Sc_0)\). Replacing \(c_0\) by \(C_0\) preserves the inequality because \(N\geq0\). The function \(v=-N\leq0\) then satisfies \[(a^{ij}\partial_i\partial_j-b^i\partial_i-C_0)v\geq0.\] Its zeroth-order coefficient is nonpositive, and a zero of \(N\) is a nonnegative interior maximum of \(v\). The strong maximum principle forces local constancy. Connectedness of \(S\) proves the dichotomy. ◻

Proof. Let \(\xi=\partial_{t_0}\). Stationarity of \(\Phi\) and \(\mathop{\mathrm{Ric}}_\gamma=2\Phi^*G\) imply \(\mathop{\mathrm{Ric}}_\gamma(\xi,\cdot)=0\). The three-dimensional Killing twist \[\mathcal W=*_\gamma \bigl(\xi^\flat_\gamma\wedge\mathrm d\xi^\flat_\gamma\bigr)\] is a scalar. Differentiating with the parallel volume tensor, using \(\nabla_i\xi_j=-\nabla_j\xi_i\) and the Killing second-derivative identity, gives \(\mathrm d\mathcal W=\pm2*_\gamma (\xi^\flat_\gamma\wedge\mathop{\mathrm{Ric}}_\gamma(\xi,\cdot))=0\). Under \(\gamma=X^2H_3\), the covector wedge acquires the factor \(X^4\) and the Hodge star on three-forms the factor \(X^{-3}\). Consequently \(\mathcal W\) is \(X\) times the corresponding twist for \(H_3\). The latter is bounded near any ordinary interior axis point: \(H_3\) is smooth and nondegenerate there because \(N>0\). Since \(X\) vanishes there, the constant \(\mathcal W\) is zero. Substituting \(\xi^\flat_{H_3}=-\mu_*\mathrm dt_0+\alpha_Y\) into Frobenius gives \[ \mu_*\,\mathrm d\alpha_Y=\mathrm d\mu_*\wedge\alpha_Y . \tag{188}\] This identity extends smoothly to the doubled meridian.

Where \(\mu_*>0\), Equation (188) makes \(-\alpha_Y/\mu_*\) closed and gives the local static form (187). In these coordinates \(\gamma=-\rho^2\mathrm dt^2+X^2h_+\). The \(tt\) Ricci equation is \(\rho\Delta_{X^2h_+}\rho=0\); conformal invariance of the two-dimensional scalar Laplacian then gives \(\Delta_{h_+}\rho=0\).

We next exclude an interior null set. First there is a positive collar of the inner boundary. The boundary alternatives from Proposition 33 are exhaustive. If \(N>0\) on \(S\), then \(Y=N\nu\), \(\operatorname{sym}\nabla Y=-Nk\), and (159) give \[\partial_\nu\mu_* =2N\bigl(\partial_\nu N+Nk(\nu,\nu)\bigr) =2N\kappa>0.\] If \(N=0\) on \(S\), then \(Y=0\) there. Its tangential derivatives vanish, and the symmetric-gradient equation makes its remaining first derivatives vanish as well. At inward boundary distance \(s\), \[Y=O(s^2),\qquad N=\kappa s+O(s^2),\qquad \mu_*=\kappa^2s^2+O(s^3)>0\quad(s>0\text{ small}).\] These collars can be chosen equivariantly and are uniform through the poles by compactness. The end also has \(\mu_*>0\), since \(N\to1\) and \(Y\to0\).

Suppose that the compact interior set \(\mathcal Z=\{\mu_*=0\}\) on the double is nonempty. Proposition 33 gives \(\mu_*\ge0\) everywhere and \(N>0\) in the interior, so \(Y\ne0\) on \(\mathcal Z\). Let \(Z\) be the \(h\)-orthogonal quarter-turn of \(Y\), using the orientation of the double. Evaluation of (188) on \((Y,Z)\) yields, on a neighborhood of \(\mathcal Z\), \[ Z(\mu_*)=-\frac{\mathrm d\alpha_Y(Y,Z)}{|Y|_h^2}\,\mu_* . \tag{189}\] The coefficient is smooth. Uniqueness for this scalar ordinary differential equation shows that the flow of \(Z\) preserves \(\mathcal Z\). A trajectory starting there exists for all time, because it stays in a compact set on which \(Z\) is smooth. Its omega-limit set has no equilibrium. Extend \(Z\) smoothly in the plane around this compact set and apply the planar Poincaré–Bendixson theorem: \(\mathcal Z\) contains a periodic orbit, a simple closed curve \(C\).

Every component of \(\{\mu_*>0\}\) on the bounded side of \(C\), intersected with either open regular half-meridian, would carry a positive harmonic function \(\rho=X\sqrt{\mu_*}\). This function extends continuously to the compact closure and vanishes on its entire frontier: the frontier lies in an axis, the inner boundary, or \(\mathcal Z\). A positive maximum would therefore occur at an interior point, where \(h_+\) is smooth and elliptic. The local strong maximum principle gives a contradiction. No ellipticity at the frontier is needed in this argument. Intersections of \(C\) with a reflected axis cause no additional frontier: first restrict to either open half and then take a component; its relative frontier is contained in the listed sets, with \(C\) itself already contained in \(\mathcal Z\).

If \(C\) surrounds the inner obstacle, the positive collar supplies such a component. Otherwise \(C\) bounds a disk disjoint from the obstacle. The same maximum argument, together with \(\mu_*\ge0\), forces \(\mu_*=0\) on that entire disk; a positive point on an axis would have positive off-axis neighbors. Hence \(Y\), and therefore \(Z\), is nonvanishing on the closed disk. Its restriction to \(C\) is a tangent field to a simple periodic orbit and has winding number one, whereas a nonvanishing field extending over a disk has winding number zero. This last contradiction proves \(\mu_*>0\) throughout the interior. Simple connectivity now gives the global primitive \(f\) in (187). ◻

Proof. In the zero-lapse case, the preceding Taylor expansions give \(\mu_*=s^2(\kappa^2+O(s))\). Taylor division on \(s\ge0\) shows that \(\lambda=s(\kappa+O(s))\), \(\alpha_Y/\mu_*\), and \(\alpha_Y^2/\mu_*\) are smooth, the last vanishing to order two. Thus \(h_+\) extends smoothly with the stated boundary value and normal derivative, and \(\mathrm df\) extends smoothly. Its primitive also extends, after fixing an additive constant. This argument only requires one-sided smoothness.

In the positive-lapse case, \(\mu_*\) itself is a boundary defining coordinate. At \(S\), \(\mathrm d\mu_*=2N\kappa\,\nu^\flat\) and \(\alpha_Y=N\nu^\flat\), so in a smooth collar \((\mu_*,z)\) we can write \[\alpha_Y=a_0\,\mathrm d\mu_*+\mu_*\eta_0,\qquad a_0=\frac1{2\kappa}+\mu_*a_1 ,\] where \(a_1\) and the tangential one-form \(\eta_0\) are smooth. Pull back by \(\mu_*=\lambda^2\). Then \[ h_+=h+\bigl(2a_0\,\mathrm d\lambda+\lambda\eta_0\bigr)^2 . \tag{191}\] The pulled-back \(h\) has even tangential and normal coefficients and odd mixed coefficients under \(\lambda\mapsto-\lambda\). The one-form in parentheses changes sign under this reflection. The metric is consequently invariant, with vanishing mixed coefficients and normal coefficient \(4a_0^2=\kappa^{-2}\) at the boundary. Its tangential block is the positive boundary block of \(h\), proving nonsingularity and the asserted derivative of \(\lambda\). Moreover \[\mathrm df=-\frac{\mathrm d\mu_*}{2\kappa\mu_*} -a_1\mathrm d\mu_*-\eta_0.\] The smooth remainder is closed, because \(\mathrm df\) is closed. Integration in the collar gives (190).

Choose all collars equivariantly under meridional axis reflection. The preceding divisions then preserve even and odd polar parity also at the poles. On an ordinary axis away from its endpoints the added term in \(h_+\) has zero transverse radial component. Thus the usual no-cone condition is retained: \(|\mathrm dX|_{h_+}=1\) on such an axis.

We make the joint corner regularity explicit. Near a pole in the positive-lapse case choose equivariant coordinates \((\mu_*,y)\), with \(y\) a signed transverse axis coordinate. The signed extension of \(X\) is \[X=yA_X(\mu_*,y^2),\qquad A_X>0.\] The coefficients \(h_{\mu_*y}\) and \(\eta_y\), where \(\eta_0=\eta_y\mathrm dy\), are odd in \(y\), while \(h_{yy}(\mu_*,0)=A_X(\mu_*,0)^2\) by the original no-cone condition. In \((\lambda,y)\), Equation (191) has components \[\begin{split} (h_+)_{\lambda\lambda}&=4\lambda^2h_{\mu_*\mu_*}+4a_0^2,\\ (h_+)_{\lambda y}&=2\lambda h_{\mu_*y}+2a_0\lambda\eta_y,\\ (h_+)_{yy}&=h_{yy}+\lambda^2\eta_y^2. \end{split}\] They extend smoothly under both commuting reflections \(\lambda\mapsto-\lambda\), \(y\mapsto-y\). At the corner the metric is diagonal and positive, with normal entry \(\kappa^{-2}\), and along \(y=0\) its \(yy\) entry is \(A_X(\lambda^2,0)^2\). Thus no-cone regularity persists jointly up to the pole. The zero-lapse case uses the original smooth boundary coordinate \(s\) and the same \(y\)-reflection, without requiring an \(s\)-reflection. ◻

Proof. Reflect the static meridian across the axes. Its conformal filling at infinity is a disk with a smooth boundary. Uniformize this disk, sending infinity to its center and the axis reflection to complex conjugation; inversion gives an exterior circular coordinate, and its logarithm gives the asserted strip. Smooth isothermal and boundary conformal regularity give nonsingular charts through the ordinary axes and up to the inner boundary, including their intersections. This uses the smooth boundary established in Lemma 41. In the positive-lapse case its two commuting corner reflections are carried to the coordinate reflections. Near \(\vartheta=0\), their smooth even and odd parities give \[ \sigma=\lambda A_s(\lambda^2,y^2),\qquad \vartheta=yA_\vartheta(\lambda^2,y^2),\qquad A_s,A_\vartheta>0 , \tag{194}\] with smooth coefficients; near the other pole use \(\pi-\vartheta\) instead. This follows by dividing each odd function by its coordinate and applying smooth even-function factorization in the two variables. It does not require analyticity. Thus \(\sigma/\lambda\), \(\sin\vartheta/y\), and \(X/\sin\vartheta\) are smooth positive ratios jointly at the corner. In the zero-lapse case the same polar conclusion follows from the smooth chart in \((s,y)\), with the \(y\)-reflection.

For clarity, the end normalization in this construction retains an initially unknown positive constant. We use the reflected isothermal expansion (55), constructed in Section 5.3. The estimate \(Y=O_2(r^{-2})\) gives \(h_+-h=O_2(r^{-4})\) on the end. After inversion of an asymptotically Euclidean meridional chart, the reflected conformal structure extends across the origin with Hölder coefficients of every exponent \(q'<1\). Its isothermal rectification has a nonzero first derivative there; rescaled annulus estimates give the differentiated remainder. Reflection supplies the same expansion in ratio at the axes. It follows that, for some \(c_\infty>0\), \[\frac{X}{e^\sigma\sin\vartheta}\longrightarrow c_\infty, \qquad \frac{\sqrt{P_+}}{e^\sigma}\longrightarrow c_\infty\] uniformly, the first quotient interpreted by its continuous polar extension. Since \(\lambda\to1\), the same first limit holds for \(\rho\). The harmonicity proved in Lemma 40 becomes flat harmonicity in the strip, and \(\rho\) vanishes continuously on each finite side.

Put \(b'=2c_\infty\). For any \(\varepsilon>0\), at a sufficiently distant line \(\sigma=L\) the functions \((b'\pm\varepsilon)\sinh\sigma\sin\vartheta\) bound \(\rho\). They have the same zero values on the finite sides. Maximum comparison on the finite rectangles, followed by \(L\to\infty\) and then \(\varepsilon\downarrow0\), gives \[\rho=b'\sinh\sigma\sin\vartheta .\] At the inner boundary, differentiation toward the domain gives \(\partial_{\nu_+}\rho=\kappa X\), because \(\rho=X\lambda\) and \(|\mathrm d\lambda|_{h_+}=\kappa\). As \(\nu_+=P_+^{-1/2}\partial_\sigma\), this says \[X\sqrt{P_+}=\frac{b'}{\kappa}\sin\vartheta \qquad(\sigma=0).\] The original and static metrics agree tangentially there. The area of the original invariant boundary is therefore \[A=2\pi\int_0^\pi X\sqrt{P_+}\,\mathrm d\vartheta =\frac{4\pi b'}{\kappa}.\] Since \(\kappa=b/R_0^2\) and \(A=4\pi R_0^2\), we obtain \(b'=b\). ◻

Proof. Let \(\Phi_*\) denote the model map in (37), with the same ordered axis constants as \(\Phi\), and set \[w=\sinh\sigma\sin\vartheta,\qquad D=\mathop{\mathrm{dist}}_G(\Phi,\Phi_*)^2.\] Both maps satisfy the harmonic-map equation with weight \(w\). The target is complete, simply connected, and nonpositively curved, as established in the axial reduction. Joint convexity of squared distance along its geodesics, applied to the two weighted harmonic-map equations, gives \[ \mathop{\mathrm{div}}(w\nabla D)\ge0 \tag{195}\] in the open strip; derivatives and divergence in this display are Euclidean.

We verify the boundary controls needed to use this inequality. By Lemma 5, on every bounded radial range the two axial lengths are comparable to the same ordinary transverse radius. Their logarithmic difference is consequently bounded. If \(y\) is that radius, the electromagnetic potential differences from their common axis values are \(O(y^2)\), and the adjusted central potential differences are \(O(y^4)\). In particular, writing \(\Delta\psi=\psi-\psi_*\), \(\Delta\chi=\chi-\chi_*\), the central displacement \[(z-z_*)+\psi_*\Delta\chi-\chi_*\Delta\psi\] is \(O(y^4)\). To bound distance, first change the height \(u=\log X\), then join the remaining three coordinates at fixed height. The target length of the latter path is controlled by \[X_*^{-1}(|\Delta\psi|+|\Delta\chi|) +X_*^{-2}|(z-z_*)+\psi_*\Delta\chi-\chi_*\Delta\psi|.\] These quantities are bounded, indeed vanish at a regular axis. Lemma 41 preserves the ordinary transverse radius by a smooth positive ratio at the horizon–axis corners, so the same argument applies there. Away from the axes smoothness of both maps gives boundedness directly. Thus \(D\) is bounded on every compact radial range, including all finite boundary faces and corners.

At infinity the matched scale in (192) gives \(u-u_*\to0\) uniformly. The end estimates (23) for the normalized potentials, including their polar ratios, give respectively \[X_*^{-1}(|\Delta\psi|+|\Delta\chi|)=O(e^{-\sigma}), \qquad X_*^{-2}|(z-z_*)+\psi_*\Delta\chi-\chi_*\Delta\psi| =O(e^{-2\sigma}).\] Here the common axis constants remove the otherwise singular polar terms. The preceding path estimate shows that \(D\to0\) uniformly as \(\sigma\to\infty\).

Fix \(\delta>0\) and set \(v=(D-\delta)_+\). It vanishes on a sufficiently distant tail and is bounded. Test (195) with \(\chi_{\mathrm{cut}}^2v\), where \(\chi_{\mathrm{cut}}\) is a compactly supported cutoff in the open strip. The weak chain rule and Cauchy’s inequality give \[ \int w\chi_{\mathrm{cut}}^2|\nabla v|^2 \le4\int wv^2|\nabla\chi_{\mathrm{cut}}|^2 . \tag{196}\] There is no outer error if the outer transition of \(\chi_{\mathrm{cut}}\) is placed in the zero tail of \(v\). At each finite side, with Euclidean distance \(x\) to that side, use a cutoff that rises from zero to one on \(\varepsilon^2<x<\varepsilon\) with derivative \((x|\log\varepsilon|)^{-1}\). On a fixed radial range \(w\le Cx\), and \[\int_{\varepsilon^2}^{\varepsilon} x\left|\frac1{x\log\varepsilon}\right|^2\mathrm dx =\frac1{|\log\varepsilon|}\longrightarrow0.\] Products of these cutoffs treat the corners; their gradient energies are bounded by the sum of the individual energies. Letting \(\varepsilon\downarrow0\) in (196) proves \(\nabla v=0\) throughout the open strip. Its zero tail makes \(v=0\). Finally \(\delta\downarrow0\) gives \(\Phi=\Phi_*\).

It remains to determine a metric coefficient; map equality alone does not yet do this. Put \[q=\log(X\sqrt{P_+}),\qquad q_*=\log(X_*\sqrt P),\] where \(P=r^2+a^2\cos^2\vartheta\) is the model coefficient. The static quotient has spatial metric \(e^{2q}(\mathrm d\sigma^2+\mathrm d\vartheta^2)\). Its scalar constraint, or directly (29), gives \(-\Delta q=|\partial\Phi|_G^2\); hence \(q-q_*\) is harmonic. It vanishes on \(\sigma=0\) by (193). At an ordinary axis the no-cone condition says \(\sqrt{P_+}=|\partial_\vartheta X|\). Since \(X=X_*\), the two nonsingular conformal scales have the same axis values. Inside the strip their logarithmic difference is precisely \[q-q_*=\frac12\log\frac{P_+}{P}.\] It extends boundedly through the corners, because the static base metrics are smooth and positive, and tends uniformly to zero at infinity: both square-root scales are asymptotic to \((b/2)e^\sigma\). Maximum comparison now gives \(q=q_*\). Thus \(P_+=P\), \(X=X_*\), and \(\lambda=\rho/X=\lambda_*\), proving the proposition. The oriented crossing order of the axes, future static time, and increasing original azimuth have been preserved throughout; the Hodge-star conventions in (28) therefore agree. ◻

Proof. Proposition 43 identifies the horizontal Lorentz metric, the map, and the map’s normal velocities on the graph \(t=f\). In particular, the curvature identity in (28), including \[X^{-2}\omega(n)=s_3=-\frac{Xf_a}{2}, \qquad \mathrm d\mathcal A=f_a\,\mathrm dA_h,\] identifies the pullback of \(\mathrm d(-\omega_*\,\mathrm dt)\) with \(\mathrm d\mathcal A\). Consequently \(\mathrm d(\mathcal A+\omega_*\,\mathrm df)=0\). The meridian is simply connected, so the real primitive \(\beta\) in (197) exists globally. On the graph the model’s fiber form becomes \[\mathrm d\phi_{\mathrm{KN}}-\omega_*\,\mathrm dt =\mathrm d\phi+\mathrm d\beta-\omega_*\,\mathrm df =\mathrm d\phi+\mathcal A.\] Its horizontal metric is \(h_+-\lambda^2\mathrm df^2=h\) by (187). The induced metric is therefore exactly the original \(g\).

We also identify the full second fundamental form. The identities (34) make this explicit. The horizontal quotient tensor, together with \(n(\log X)\), determines the horizontal block \(K|_h\), by the tensor relation used in Proposition 33. The fiber-fiber unit entry is \(K(e,e)=n(\log X)\). Finally the spatial identity \[\omega=X^2*_h l,\qquad l=K(e,\cdot)|_h,\] determines the mixed block. All these quantities agree under the map, with the same future normal, so no component of \(K\) is discarded in undoing polarization.

Likewise, \(\mathrm d\psi=X*_hE'_H\) and \(\mathrm d\chi=X*_hB'_H\) recover both horizontal electromagnetic components. Their normal map velocities recover the remaining components through \(E'_e=s_2\), \(B'_e=-s_1\). Thus the rotated Maxwell fields agree. If \(Q>0\), apply \[ E_f=\frac{Q_eE'-Q_bB'}Q,\qquad B_f=\frac{Q_bE'+Q_eB'}Q; \tag{198}\] if \(Q=0\), use the identity rotation. The model then has the ordered original charges \((Q_e,Q_b)\). The constant rotation preserves the stress tensor and \(E_f\times B_f\), so it also preserves the total angular-momentum normalization. The model choice \(a=-J/M\) consequently has exactly the prescribed \(J\) with the positive convention for \(K\).

Every interior meridional change above respects axis reflection. The differential defining the angular rotation has ordinary invariant polar parity, so its primitive gives a smooth rotation of each normal polar disk. This extends the map through the axes. Finally \(\mathrm df=-\alpha_Y/\mu_*=O(r^{-2})\), and the global static meridional coordinates tend to the model’s spatial infinity. ◻

Proof. In the zero-lapse case the static meridional coordinates and \(f\) are already smooth up to \(S\). In the positive-lapse case, the isometric reflection in Lemma 41, followed by conformal reflection across the inner circle, shows that \(\sigma\) is odd and \(\vartheta\) even as functions of \((\lambda,z)\). Hence \[\frac{\sigma}{\lambda}>0,\qquad \vartheta\] extend as smooth even functions, and therefore as smooth one-sided functions of the original coordinate \(\mu_*=\lambda^2\). At the poles, the joint smooth positive ratios in (194) give the original polar extension.

The apparent singularity of the angular variable is removed in co-rotating coordinates. Put \[\omega_H=\frac a{R_0^2},\qquad \phi_\sharp=\phi_{\mathrm{KN}}-\omega_H t.\] On the graph its angular adjustment has differential \[\mathrm d\beta-\omega_H\mathrm df =\mathcal A+(\omega_*-\omega_H)\mathrm df.\] The model expansion is \(\omega_*-\omega_H=\sigma^2\omega_2(\sigma^2,\vartheta)\), with smooth polar coefficients. This differential is therefore smooth also in the positive-lapse case, where \(\mathrm df=-\mathrm d\mu_*/(2\kappa\mu_*)+\mathrm df_0\). It is closed and has a smooth primitive up to the boundary, with the same axis parity as before.

Use local horizon coordinates \[ U=-\sigma e^{-\kappa t},\qquad V=\sigma e^{\kappa t}. \tag{199}\] They satisfy the exact identities \[ -UV=\sigma^2,\qquad -\mathrm dU\,\mathrm dV=\mathrm d\sigma^2-\kappa^2\sigma^2\mathrm dt^2,\qquad \sigma^2\mathrm dt=\frac{V\mathrm dU-U\mathrm dV}{2\kappa}. \tag{200}\] To check that these are smooth nonsingular extension coordinates, the model has \[\lambda_*^2=P\kappa^2\sigma^2 +\sigma^4 L(\sigma^2,\vartheta),\] where \(L\) and all the remaining model coefficients are smooth in \(\sigma^2\) and the ordinary angular variables. The leading horizontal block is \[P(\mathrm d\sigma^2-\kappa^2\sigma^2\mathrm dt^2) =-P\,\mathrm dU\,\mathrm dV.\] The error \(\sigma^4L\mathrm dt^2\) is a smooth quadratic expression in \(V\mathrm dU-U\mathrm dV\). The fiber form in co-rotating coordinates is \(\mathrm d\phi_\sharp-(\omega_*-\omega_H)\mathrm dt\), which is smooth by (200). Thus the complete Lorentz metric extends nonsingularly, using ordinary angular patches at the poles. For the electric model potential, its coefficient on the horizon generator is constant. Subtract that constant times \(\mathrm dt\); the remaining time coefficient is \(O(\sigma^2)\), smooth in \(\sigma^2\), so its field also extends smoothly. The same holds after the constant duality rotation.

On the graph, if \(N|_S=0\), both \(U\) and \(V\) vanish smoothly at \(S\). This is the bifurcation sphere. If \(N|_S>0\), use (190) to write instead \[ V=\frac{\sigma}{\sqrt{\mu_*}}e^{\kappa f_0},\qquad U=-\frac{\sigma}{\sqrt{\mu_*}}\,\mu_*e^{-\kappa f_0}. \tag{201}\] Both are smooth in the original variables, with \(U=0\), \(V>0\) on \(S\). This is the future horizon branch: in the exterior \(U<0<V\), and the future stationary generator has components \(\kappa(-U\partial_U+V\partial_V)\). Figure 1 summarizes these two attachments.

The extended map preserves the smooth positive original metric. It is thus an immersion and is spacelike up to \(S\); the future unit normal then extends smoothly as well. It is injective in the interior, since its static meridional coordinates form a global coordinate change and each fiber map is a rotation. On the boundary the map to the horizon cross-section coordinates is a diffeomorphism, so distinct boundary points remain distinct. No interior point maps to the horizon. Finally compactness on bounded radial sets and properness at the end make the injective map proper onto its image, hence an embedding. The induced \(K,E_f,B_f\), already identified in the interior, agree at \(S\) by smooth extension. ◻

Proof. Write \(r_x=|x|\) in the prescribed slice chart, let \(T\) be the stationary time Killing field future at infinity, and decompose \[T=N_Tn+Y_T .\] Since the end approaches ambient spatial infinity, \(N_T>|Y_T|_g\) eventually and \(N_T^2-|Y_T|_g^2=-\bar g(T,T)\to1\). We first obtain \[ N_T\to1,\qquad Y_T=O_2(r_x^{-2}). \tag{203}\] No prior bound on the tilt of the embedding is needed. The Gauss equation bounds the tangential ambient curvature components in the slice frame by \(O(r_x^{-3})\); Codazzi bounds the components with one normal index by the same order. The remaining normal-normal block follows by contraction with the ambient Ricci tensor, whose components in this frame are \(O(r_x^{-4})\) by the given electromagnetic fields and the Einstein–Maxwell equation. Thus \(\overline{\operatorname{Rm}}=O(r_x^{-3})\) in the entire slice frame.

The Killing first-jet equation \(\bar\nabla^2T=\overline{\operatorname{Rm}}*T\), split into tangential and normal components, therefore has the end bounds used in Lemma 38. The splitting adds only \(K*\bar\nabla T\) and \((\nabla K+K*K)*T\); the skew symmetry of \(\bar\nabla T\) controls all its components by the tangential derivatives. In slice coordinates the Christoffel symbols and their derivatives are \(O(r_x^{-2})\), \(O(r_x^{-3})\). The radial first-jet integration and causality argument of that lemma consequently give constant limits with \(O_2(r_x^{-q'})\) errors, for any fixed \(1/2<q'<1\). Here the normalization is determined by \(-\bar g(T,T)\to1\), rather than by a variational prototype.

For the original unpolarized data, the constraints and prescribed falloff give \[L(g-\delta)=\partial_i\partial_j(g-\delta)_{ij} -\Delta_\delta\mathop{\mathrm{tr}}_\delta(g-\delta)=O(r_x^{-4}).\] The pair \((N_T,Y_T)\) satisfies \(\operatorname{sym}\nabla Y_T=-N_TK\), with \(K=O_1(r_x^{-3})\), bounded \(N_T\), and the constant asymptotics just proved. Since \(m>0\), the general assertion of Lemma 38 gives \(Y_{T,\infty}=0\). The norm limit then gives \(N_{T,\infty}=1\). Prolonging \(\operatorname{sym}\nabla Y_T=-N_TK\) gives \[|\partial^2Y_T| \le C r_x^{-2}|\partial Y_T| +Cr_x^{-3}|Y_T|+Cr_x^{-4}.\] Twice integrating from the zero end limits, and reinserting the improved bounds, yields (203).

Project along \(T\) to stationary spatial coordinates \(\bar x\) of the ambient exterior, and restrict to \(\bar r=|\bar x|\) sufficiently large. This projection is a local diffeomorphism on the slice, because a timelike vector cannot be tangent to a spacelike hypersurface. It is proper over that spatial tail: the end goes to ambient spatial infinity, and a compact core has bounded projected radius. Explicitly, for every ambient radius \(R\) there is a slice radius \(R_x(R)\) such that \[|x|>R_x(R)\quad\Longrightarrow\quad |\bar x|>R.\] Thus the preimage of the ambient tail contains one whole connected intrinsic coordinate tail, and the preimage of any compact subset of the ambient tail is a closed subset of an intrinsic compact set. Its image is therefore open and closed in the connected tail, so it is surjective and a covering. The spatial tail is simply connected. Each component of its preimage covers the whole tail and hence reaches arbitrarily far out in the slice end: otherwise its closure would lie in an intrinsic compact set, whose projected radius is bounded. All such components must therefore intersect the one connected full intrinsic tail just specified. There is just one component and one sheet. This proves the single-graph assertion.

The stationary orbit metric on this three-dimensional end pulls back to \[ g_T=g+\frac{(Y_T^\flat)^2}{N_T^2-|Y_T|^2}. \tag{204}\] By (203), it differs from \(g\) by \(O_2(r_x^{-4})\) and has the same mass. In the ambient stationary chart, the same metric differs from the canonical spatial metric by \(\bar g_{ti}\bar g_{tj}/(-\bar g_{tt})=O_2(\bar r^{-4})\). The canonical mass is \(M\).

Here the equality of masses in the two charts can be checked directly. Since both represent the same uniformly Euclidean metric, \(D\bar x\) and its inverse are bounded. Distances in the two ends show that \(r_x\) and \(\bar r\) are comparable. The transformation law for the Christoffel symbols then gives \(D^2\bar x=O(r_x^{-2})\). Integrating along rays and sphere paths yields a constant orthogonal matrix \(O_*\) with \[ D\bar x=O_*+O(r_x^{-1}),\qquad \bar x=O_*x+O(\log r_x). \tag{205}\] The ADM flux comparison separates a pure Euclidean symmetric gradient, whose linear flux vanishes, from the original perturbation. Quadratic errors tend to zero. The Euclidean scalar linearization is integrable (\(O(r_x^{-4})\)), so asymptotically round deformed spheres give the same limit as round spheres. Thus \(m=M\).

In the \(\bar x\)-chart the graph satisfies \[ \mathrm dF= \frac{\bar g_{ti}\mathrm d\bar x^i-Y_T^\flat} {N_T^2-|Y_T|^2} =O_1(\bar r^{-2}). \tag{206}\] Its vanishing tilt leaves the leading Coulomb fluxes unchanged, giving \(Q_e^2+Q_b^2=\widetilde Q_e^2+\widetilde Q_b^2\); possible orientation changes do not affect this identity. The graph tangents and the normalized future normal covector proportional to \(-\mathrm dt+\mathrm dF\) give, with the positive convention for \(K\), \[ K_{ij}=K^{\rm can}_{ij}+\partial_i\partial_jF +O(\bar r^{-4}). \tag{207}\] For every Euclidean rotation vector \(R\), \[n_\delta^i(\partial_i\partial_jF)R^j =R\left(\partial_{\bar r}F-\frac{F}{\bar r}\right).\] Its round-sphere integral is zero because \(R\) preserves the sphere area form. The trace-reversed Euclidean term also vanishes, since \(R\) is tangential; all metric, normal, and area errors tend to zero at these decay orders. Hence the angular flux vector in this chart has the canonical norm \(|M\widetilde a|\).

Under (205), rotation vectors agree, up to the constant orthogonal change, with errors \(O(\log r_x)\). Their flux errors are \(O(\log r_x/r_x)\), since \(K=O_1(r_x^{-3})\). The corresponding spheres differ in radius by \(O(\log r_x)\), and their normals and relative area elements by \(O(\log r_x/r_x)\); these also give vanishing angular-flux errors. In the prescribed slice chart, axisymmetry forces the two transverse components of the angular flux vector to vanish. Its axial component is the prescribed total \(J\), because the electromagnetic correction tends to zero at infinity. Thus its norm is \(|J|\), proving \(J^2=M^2\widetilde a^2\). ◻

Proof. This is a direct boundary argument, independent of the adjoint boundary dichotomy. In the smooth co-rotating coordinates (199), write the section as \(U=0,\ V=v(x)\ge0\), where \(x\) ranges over the sphere of generators. Such a global graph exists because generator projection is a diffeomorphism. Smoothness of the model coefficients in \(-UV\), with the weights imposed by the horizon Killing field \(\kappa(-U\partial_U+V\partial_V)\), gives on \(U=0\) \[\bar g_{UV}=-c,\quad \bar g_{UA}=Vb_A,\quad \bar g_{AB}=q_{AB},\quad \bar g_{VA}=\bar g_{VV}=0,\qquad c>0,\] and \[\partial_U\bar g_{AB}=Vd_{AB},\qquad \partial_U\bar g_{VA}=e_A,\qquad \partial_U\bar g_{VV}=0.\] Here \(q\) is a positive sphere metric and all coefficients are smooth functions of \(x\). The last identity follows from the stronger form \(\bar g_{VV}=U^2a_2(UV,x)\); merely knowing \(\bar g_{VV}=0\) on the horizon would not suffice.

Set \(\ell=\partial_V\), and choose the transverse future null normal \(k\) with \(\bar g(k,\ell)=-1\). The section tangents are \(e_A^{\,S}=\partial_A+v_A\partial_V\); orthogonality gives \[k^U=\frac1c,\qquad k^A=q^{AB}\left(v_B-\frac{v b_B}{c}\right),\qquad k_V=-1,\qquad k_A=v_A.\] The remaining component is fixed by nullness and is immaterial for the expansion because the horizon’s outgoing second fundamental form is zero. Direct contraction of \(-\bar g(k,\bar\nabla_{e_A^{\,S}}e_B^{\,S})\) with \(q^{AB}\) gives the linear expression \[ \theta_k[v]= \Delta_q v+ \frac1c\langle\mathrm dc+e-b,\mathrm dv\rangle_q+ \frac{\mathop{\mathrm{tr}}_qd-2\operatorname{div}_q b}{2c}\,v . \tag{208}\] For example the relevant lower Christoffel coefficients are \[\Gamma_{U;AB} =\frac V2(\partial_A b_B+\partial_B b_A-d_{AB}),\qquad \Gamma_{U;VB} =\frac12(b_B-\partial_Bc-e_B);\] combining these with \(k_V=-1\), \(k_A=v_A\) yields (208). The \(VV\) coefficients vanish on the horizon, so there is no quadratic gradient term.

Outer area minimization, applied to arbitrary nonnegative outward variations, implies \(H\ge0\). Since \(\theta_+=H+\mathop{\mathrm{tr}}_SK=0\), the other future expansion is \(\theta_-=\mathop{\mathrm{tr}}_SK-H=-2H\le0\). The outward future normal \(n+\nu\) is a positive multiple of \(\ell\), so \(n-\nu\) is a positive multiple of \(k\). Consequently \(\theta_k[v]\le0\). Equation (208) is a uniformly elliptic linear inequality for the nonnegative smooth function \(v\). The strong minimum principle applies even if its zeroth-order coefficient has either sign: in the operator \(-\theta_k\), replace that coefficient by its positive part, which preserves the supersolution inequality because \(v\ge0\). Hence a zero of \(v\) forces \(v\equiv0\) on the connected sphere. Otherwise \(v>0\) everywhere, proving the two alternatives. ◻

Completion of Theorem 2. Corollary 19 proves the numerical inequality on the closed physical branch. On the strict branch, Proposition 33 supplies the stationary quotient and its boundary conditions. Lemmas 40–42 and Proposition 43 identify that quotient with the subextremal model determined by Lemma 9. Propositions 44 and 45 give the required smooth embedding of the original metric, tensor, and electromagnetic fields, with its future normal and its stated boundary image. The horizon area formula and Lemma 46 prove the converse, and the canonical examples verify sharpness within the stated class. No classification on the extremal equality branch is asserted. ◻