Proof. Elliptic interior estimates, applied on unit balls, give smoothness and uniform bounds for every fixed derivative order of \(u\). The function \(h=u_8>0\) satisfies \((-\Delta+3u^2-1)h=0\). For every compactly supported \(C^1\) function \(\eta\), integration by parts gives the ground-state identity \[ \int_{\mathbb R^8}\bigl(|\nabla\eta|^2+(3u^2-1)\eta^2\bigr) =\int_{\mathbb R^8}h^2\left|\nabla\left(\frac\eta h\right)\right|^2 \ge0. \tag{9}\] The division is legitimate because \(h\) has a positive minimum on the support of \(\eta\).

Bounded monotonicity gives pointwise limits along each vertical line. Translation compactness and the derivative bounds improve this to (8). Indeed every convergent subsequence has the same pointwise limit, and a bounded vertical shift leaves that limit unchanged. Passing the equation and stability to these limits first gives solutions and stability in eight dimensions. To see stability in seven dimensions, fix \(\varphi\in C_c^1(\mathbb R^7)\) and a nonzero \(\zeta\in C_c^1(\mathbb R)\). Test with \(\eta(x',t)=\varphi(x')\zeta(t/L)\) and divide by \(L\int\zeta^2\). The result is \[\int_{\mathbb R^7}\bigl(|\nabla\varphi|^2+(3v_\pm^2-1)\varphi^2\bigr) \ge -\frac1{L^2}\frac{\int|\zeta'|^2}{\int\zeta^2} \int_{\mathbb R^7}\varphi^2.\] Letting \(L\to\infty\) proves the assertion. For example, \(u(x',t)<u(x',t+1)\le v_+(x')\), which proves the strict upper inequality; the lower one is analogous.

Theorem 2 says that each \(v_\pm\) is a constant well or a planar heteroclinic. To check the minimizing assertion directly, put \(\Phi(a)=\int_{-1}^a\sqrt{2W(b)}\,\mathrm db\). For a unit vector \(e\) and a competitor \(w\in H^1(\Omega,[-1,1])\), \[ \frac12|\nabla w|^2+W(w) \ge \sqrt{2W(w)}\,\partial_e w =\partial_e\Phi(w). \tag{10}\] There is equality when \(w(x)=g(e\cdot x-c)\), since \(g'=\sqrt{2W(g)}\). The integral of the last expression depends only on the boundary trace. Thus such a profile minimizes on every ball, in any ambient dimension in which \(e\) is a unit vector. This includes the extensions of the seven-dimensional profiles, with \(e_8=0\). The constants \(\pm1\) have zero energy and also minimize. ◻

Proof. Fix a ball \(B\). The direct method produces an energy minimizer \(w\) with boundary trace \(u\); truncation allows \(|w|\le1\). First clip it from below by \(a_-\), writing \(w_1=\max\{w,a_-\}\). The Sobolev lattice identity is \[\mathcal E(\max\{w,a_-\};B) +\mathcal E(\min\{w,a_-\};B) =\mathcal E(w;B)+\mathcal E(a_-;B).\] Since \(u>a_-\) on \(\partial B\), the minimum on the left has the same trace as \(a_-\). Minimality of \(a_-\) therefore shows \(\mathcal E(w_1;B)\le\mathcal E(w;B)\). The maximum has trace \(u\), so it remains an unconstrained minimizer. The analogous upper clipping gives another unconstrained minimizer \(\widetilde w=\min\{w_1,a_+\}\) satisfying \(a_-\le\widetilde w\le a_+\).

The Euler–Lagrange equation and elliptic boundary regularity make \(\widetilde w\) smooth up to \(\partial B\). A difference of two solutions \(b,c\) here satisfies \(-\Delta(b-c)+(b^2+bc+c^2-1)(b-c)=0\), with bounded zeroth-order coefficient. Strong comparison for a nonnegative solution of this linear equation, together with the strict boundary inequalities, gives \[a_-<\widetilde w<a_+\quad\text{on }\overline B.\] These gaps have positive minima on this compact set. Vertical translates \(u_s(x',t)=u(x',t+s)\) converge uniformly there to \(a_+\) as \(s\to+\infty\), so \(u_s>\widetilde w\) for all sufficiently large \(s\). Let \[s_* =\inf\{s\ge0:u_s\ge\widetilde w\text{ on }\overline B\}.\] The set inside the infimum is an upper interval, by strict monotonicity. Continuity gives \(u_{s_*}\ge\widetilde w\). If \(s_*>0\), then \(u_{s_*}>u=\widetilde w\) on \(\partial B\), and strong comparison gives strict inequality throughout \(\overline B\). Compactness and continuity then allow a smaller positive translation, a contradiction. Thus \(\widetilde w\le u\). For the reverse inequality, \(u_{-s}\) lies below \(\widetilde w\) when \(s\) is large. Decrease \(s\ge0\) until first contact. A positive first-contact parameter is again impossible, because \(u_{-s}<u=\widetilde w\) on \(\partial B\) and strong comparison gives a strict compact gap. At \(s=0\) this yields \(u\le\widetilde w\). Consequently \(u=\widetilde w\) on \(B\), and \(B\) was arbitrary.

Finally choose a smooth cutoff \(\eta_R\) that is zero on \(B_{R-1}(y)\), one near \(\partial B_R(y)\), and satisfies \(|\nabla\eta_R|\le C\). The competitor \(1+\eta_R(u-1)\) has zero energy in the inner ball and bounded energy density in the remaining shell. The latter assertion uses \(|u|\le1\) and the uniform unit-scale interior gradient estimate for the equation. The shell has volume at most \(C_NR^{N-1}\), proving (11). ◻

Proof. Compactness and the lower bound. Recall the phase primitive \(\Phi(a)=\int_{-1}^a\sqrt{2W(b)}\,\,\mathrm db\). Denote the rescaled energy by \(\mathcal E_j\) and its energy measure by \(\mu_j\). Scaling the minimizing comparison shows that \(u_j\) minimizes \(\mathcal E_j\) against every compact modification. A change of variables gives \[ \mu_j(B_r)=R_j^{1-N}\mathcal E(u;B_{R_jr}). \tag{14}\] Thus these measures are locally bounded. Moreover \[|\nabla\Phi(u_j)| \le \frac{\varepsilon_j}2|\nabla u_j|^2+\frac{W(u_j)}{\varepsilon_j}, \qquad \int_K W(u_j)\le\varepsilon_j\mu_j(K)\longrightarrow0\] on each compact set \(K\). Local BV compactness, followed by a diagonal subsequence, implies that \(\Phi(u_j)\) has an \(L^1_{\mathrm{loc}}\) limit. The vanishing potential forces its values to lie in \(\{\Phi(-1),\Phi(1)\}=\{0,s_0\}\). The inverse of \(\Phi\) is uniformly continuous on \([0,s_0]\), so this also proves the asserted local \(L^1\) convergence of \(u_j\) to \(2\mathbf1_E-1\). Weak compactness of \(\mu_j\) and lower semicontinuity of total variation give a local measure limit \(\mu\) satisfying \[ \mu\ge s_0|D\mathbf1_E|. \tag{15}\]

A recovery comparison with the correct boundary trace. We prove the opposite inequality, including the boundary gluing. Fix a ball \(B_r(y)\) and \(\delta>0\). Let \(F\) be a finite-perimeter set in \(B_{r+\delta}(y)\) that agrees with \(E\) on the annulus \(A=B_{r+\delta}(y)\setminus\overline{B_r(y)}\). The case \(F=E\) is permitted. Choose a recovery sequence \(w_j\) on this larger ball as in (12). Both \(w_j\) and \(u_j\) converge on \(A\) to \(2\mathbf1_E-1\). Their boundedness implies \[ \int_A|u_j-w_j|^2\le2\int_A|u_j-w_j|\longrightarrow0. \tag{16}\]

For \(\varepsilon_j<\delta/4\), partition all but a remainder of \(A\) into \(L_j=\lfloor\delta/\varepsilon_j\rfloor-1\) disjoint radial layers of thickness \(\varepsilon_j\). Thus \(L_j\ge\delta/(2\varepsilon_j)\), and the outer remainder leaves a neighborhood of \(\partial B_{r+\delta}(y)\) untouched. In one such layer \(S\), let \(\chi\) increase from zero to one, with \(|\nabla\chi|\le C/\varepsilon_j\), and interpolate by \(z_j=(1-\chi)w_j+\chi u_j\). The cutoff is zero on the inner side and one on the outer side. On \([-1,1]\), boundedness of \(W''\) and \(|W'|^2\le CW\) give, uniformly for \(0\le\chi\le1\), \[W((1-\chi)a+\chi b) \le C\bigl(W(a)+W(b)+|a-b|^2\bigr).\] Expanding the gradient therefore yields \[ \mathcal E_j(z_j;S) \le C\bigl(\mathcal E_j(u_j;S)+\mathcal E_j(w_j;S)\bigr) +\frac C{\varepsilon_j}\int_S|u_j-w_j|^2. \tag{17}\] Averaging the right-hand side over the disjoint layers selects \(S_j\) with cost at most \[ C\frac{\varepsilon_j}{\delta} \bigl(\mathcal E_j(u_j;A)+\mathcal E_j(w_j;A)\bigr) +\frac C\delta\int_A|u_j-w_j|^2=o_j(1). \tag{18}\] Here \(\delta\) is fixed before \(j\to\infty\); the two energies are bounded by compactness and recovery. In particular no rate for the \(L^1\) convergence is needed: the averaging cancels the factor \(\varepsilon_j^{-1}\) multiplying the mismatch in a single layer.

Use \(w_j\) inside \(S_j\), the interpolation on \(S_j\), and \(u_j\) outside \(S_j\). This is an admissible compact modification of \(u_j\). Its minimizing property, and nonnegativity of the original energy in the part of the annulus inside \(S_j\), give \[\mathcal E_j(u_j;B_r(y)) \le \mathcal E_j(w_j;B_{r+\delta}(y))+o_j(1).\] Consequently, whenever \(\mu(\partial B_r(y))=0\), \[ \mu(B_r(y))\le s_0 P(F;B_{r+\delta}(y)). \tag{19}\] Equality of energy and perimeter, then minimality. First set \(F=E\). At radii with \(\mu(\partial B_r(y))=|D\mathbf1_E|(\partial B_r(y))=0\), send \(\delta\downarrow0\) in (19). These are all but countably many radii at each center. Together with (15), the resulting inequalities on balls prove \(\mu=s_0|D\mathbf1_E|\) as measures. More explicitly, \(\mu-s_0|D\mathbf1_E|\) is already a nonnegative Radon measure by (15), and it vanishes on these balls, which form a neighborhood basis. In particular no additional transition-layer mass remains.

Next suppose \(F\mathbin\triangle E\) is compactly contained in an open ball. Choose \(r\) outside that compact set, inside the ball, and with zero perimeter measure on \(\partial B_r(y)\). Then \(F=E\) near this sphere, so it is a continuity sphere for both perimeter measures. Apply (19) and let \(\delta\downarrow0\) to obtain \(P(E;B_r(y))\le P(F;B_r(y))\). The perimeter measures coincide outside the compact modification; hence this proves local perimeter minimality, as asserted. ◻

Proof. This is the classical directed-cone reduction; see also Schnürer (2003, Theorem 4.5 and Corollary 4.6). We give its short proof to make the treatment of the vertex explicit. The cone has a smooth compact link. The singular-set dimension bound is at most zero. A nonzero singular point would, by dilation invariance, generate a whole singular ray: each dilation is an ambient diffeomorphism preserving \(C\), and therefore preserves the distinction between smooth and singular boundary points. This is impossible, so \(\operatorname{Sing}C\subset\{0\}\). Consequently \[\Lambda=C\cap\mathbb S^7\] is a smooth compact embedded minimal hypersurface of \(\mathbb S^7\), possibly with several connected components. The oriented normal is continuous on the regular boundary. Since tangent normal lines are dilation invariant and the choice of side is continuous, that orientation is constant along every regular ray: it takes values in the same two-point set along that connected ray. This uses conicality of the support and does not presume conicality of \(E\). The function \(f=\nu_E\cdot e_8\ge0\) is therefore homogeneous of degree zero.

The normal component of a Euclidean translation on a minimal hypersurface satisfies the Jacobi equation (Simons 1968, Corollary 3.3.1, p. 74) \[\Delta_C f+|A_C|^2f=0.\] The cone metric is \(\,\mathrm dr^2+r^2g_\Lambda\) and \(|A_C|^2=r^{-2}|A_\Lambda|^2\), where \(A_\Lambda\) is the second fundamental form of \(\Lambda\) in the sphere. Thus \[ \Delta_\Lambda f+|A_\Lambda|^2f=0. \tag{22}\] We separate the case where this nonnegative function is positive on a link component from the case where it vanishes on every component. A positive directional normal component. On a connected component where \(f\) is not identically zero, the strong maximum principle gives \(f>0\). Integration of (22) on that compact component yields \(\int|A_\Lambda|^2f=0\), so it is totally geodesic. A connected compact embedded totally geodesic hypersurface of the sphere is a whole equatorial \(\mathbb S^6\). Any other component, being disjoint from this equator, would lie in one of its open hemispheres. This cannot happen: the corresponding signed coordinate function \(\ell\) on a closed minimal six-dimensional hypersurface of \(\mathbb S^7\) satisfies \(\Delta_\Lambda\ell+6\ell=0\), hence has integral zero, whereas in an open hemisphere it has strict constant sign. There are consequently no other components, and \(C\) is the hyperplane over that equator.

A vanishing directional normal component. It remains to handle the case \(f\equiv0\) on all of \(\Lambda\). Perimeter measure gives zero mass to the possible singular point, so now \[ \partial_8\mathbf1_E=0\quad\text{on all of }\mathbb R^8. \tag{23}\] A locally integrable function whose distributional derivative in one direction vanishes is invariant under translations in that direction. For example this follows by mollifying, applying the assertion to the smooth functions, and passing to the local \(L^1\) limit. Thus each vertical translation preserves \(\mathbf1_E\) as an \(L^1_{\rm loc}\) class, and also preserves its perimeter measure, its support \(C\), and the singular set of that support. If \(0\) were singular, every point of the vertical axis would be singular, contradicting \(\operatorname{Sing}C\subset\{0\}\). Therefore the origin is regular. A cone smooth at its vertex equals its tangent hyperplane there: dilating toward the origin keeps the cone fixed while its smooth local graphs converge to that tangent plane. This proves flatness in the remaining case. In either case the weight is \(\mathcal H^7\mathbin\llcorner C\), by the multiplicity-one structure of a minimizing boundary. ◻

Proof. We first prove the pointwise bounds on a fixed ball, then the exponential comparison, and finally the normalized derivative estimates.

Fixed-ball control. The drift and potential in (32), including the derivatives of their coefficients, are uniformly bounded. Apply the weak Harnack inequality to the nonnegative supersolution \(\mathcal LD\ge0\) on a fixed enclosing ball, for example \(B_{20}(z)\). The constant leading matrix and bounded lower-order coefficients permit an exponent greater than one; Hölder’s inequality therefore gives \[\int_{B_4(z)}D\le C D(z).\] The enclosing radius is fixed, so the constant depends only on dimension. Integrating (32) against a nonnegative cutoff equal to one on \(B_3(z)\) and supported in \(B_4(z)\) gives \[ \int_{B_3(z)}|T|^2\le C D(z). \tag{39}\] Indeed all derivatives can be moved onto the cutoff and the bounded drift; in particular \(\operatorname{div}(vp)\) is uniformly bounded.

Put \(b=2\sqrt2 vp\). Twice differentiating (31), and using \(D_i=-2(Tp)_i\) and \(D_{ij}=-2p_k\partial_kT_{ij}-2(T^2)_{ij}\), gives the explicit system \[ (\Delta-b\cdot\nabla)T_{ij} =-qDT_{ij}+\sqrt2 vqD p_i p_j -q(p_iD_j+p_jD_i)+2\sqrt2 v(T^2)_{ij}. \tag{40}\] Repeated indices are summed. Its right side is bounded in norm by \(C(D+|T|)\), using \(\nabla D=-2Tp\) and the preliminary uniform bound on \(T\). Consequently \(F_0=D^2+|T|^2\) satisfies \[(-\Delta+b\cdot\nabla)F_0\le C F_0.\] The local mean-value estimate, (39), and \(D^2\le D\) give \(|T(z)|^2\le C D(z)\). We may now regard the \(D\) equation as a homogeneous linear equation with bounded potential \(2q|p|^2-2|T|^2/D\). Ordinary Harnack gives uniform local comparability of \(D\). Finally apply interior \(W^{2,r}\) estimates, \(r>n\), to (32): its source \(2|T|^2\) is bounded on a fixed ball by \(CD(z)\). Sobolev embedding gives \(|\nabla D(z)|\le CD(z)\). This argument uses only the boundedness of \(|T|^2/D\), not a derivative bound for that quotient.

The exponential rate. Fix \(\kappa=2\sqrt2+\gamma\). For \(r=|x-y|>0\), \[\mathcal L(e^{-\kappa r})\le e^{-\kappa r} \left[-\kappa^2+\frac{(n-1)\kappa}{r} +2\sqrt2\kappa|v|+2(1-v^2)\right].\] The maximum of the last two terms, for \(|v|\in[0,1]\), is \(2\sqrt2\kappa<\kappa^2\). Hence this is negative for \(r\ge R_\gamma\), where \(R_\gamma\) is uniform. Local Harnack, chained a fixed number of times, provides a uniform \(a_\gamma>0\) such that \(a_\gamma D(y)e^{-\kappa r}\le D(x)\) on \(r=R_\gamma\). For any \(\epsilon>0\) subtract \(\epsilon\) from the left side; this preserves the subsolution inequality since the potential of \(\mathcal L\) is nonnegative. It is then negative on a sufficiently large outer sphere. Comparison on annuli and \(\epsilon\downarrow0\) show \(D(x)\ge a_\gamma D(y)e^{-\kappa|x-y|}\) outside the inner ball. Harnack covers the inner ball and interchange of \(x,y\) gives the last bound in (36).

Normalized derivatives. Fix \(z\), put \(\epsilon=D(z)\), and set \(d_\epsilon=D/\epsilon\), \(t_\epsilon=T/\sqrt\epsilon\). The estimates already proved bound \(d_\epsilon\), \(\nabla d_\epsilon\) and \(t_\epsilon\) on each fixed ball. Their equations are \[\begin{align*} (\Delta-b\cdot\nabla)d_\epsilon &=2q|p|^2d_\epsilon-2|t_\epsilon|^2,\\ (\Delta-b\cdot\nabla)(t_\epsilon)_{ij} &=-q\epsilon d_\epsilon(t_\epsilon)_{ij} +\sqrt{2\epsilon}\,vq d_\epsilon p_i p_j -q\sqrt\epsilon\big(p_i\partial_jd_\epsilon +p_j\partial_id_\epsilon\big) +2\sqrt{2\epsilon}\,v(t_\epsilon^2)_{ij}. \end{align*}\] Their coefficients have uniform smooth bounds, and \(0<\epsilon\le1\). First use interior \(W^{2,r}\) estimates on both equations and then Schauder estimates, shrinking the ball at each step. Induction gives (37). The lower bound for \(D\) follows from the local Harnack inequality already established. Finally apply (37) on a unit ball centered at \(x\), so that \(|\nabla^kD(x)|\le C_kD(x)\) and \(|\nabla^kT(x)|\le C_k\sqrt{D(x)}\), and use the exponential comparison between \(x\) and \(z\). This proves (38); in particular the constants on logarithmically growing windows need not be inferred from an unspecified dependence of \(C_{R,k}\) on \(R\). ◻

Proof. Clearing on large balls. Let \(\phi_R\) be the positive first Dirichlet eigenfunction of a ball \(B_R(x)\), normalized by \(\phi_R(x)=\max\phi_R=1\). For \(0<c<1\) such that \(\lambda_1(B_R)<1-c^2\), \[\Delta(c\phi_R)+c\phi_R-(c\phi_R)^3 =c\phi_R\big(1-\lambda_1(B_R)-c^2\phi_R^2\big)>0.\] If \(v>0\) on a neighborhood of the closed ball, small multiples of \(\phi_R\) lie below \(v\). Increase the multiple up to \(c\). A first interior contact contradicts the strict displayed inequality at a minimum of \(v-c\phi_R\); boundary contact is excluded by \(v>0\) there. Thus \(v(x)\ge c\). Since \(\lambda_1(B_R)=R^{-2}\lambda_1(B_1)\), for each \(c<1\) this holds once \(R\) is sufficiently large. If an entire \(|v|<1\) solution had no zero, connectedness would give one strict sign; letting \(R\to\infty\) and then \(c\uparrow1\) would give \(|v(x)|\ge1\), a contradiction. The same comparison also shows, uniformly, that \(|v|\to1\) when distance to a phase boundary tends to infinity.

Decay at every rate below \(\sqrt2\). It suffices to treat a positive phase. Write \(w=1-v\); then \(0\le w\le1\) and \[\Delta w=(2-3w+w^2)w.\] Fix \(k=\sqrt2-\gamma/2\). By the preceding comparison there is a uniform \(R_0\) such that \(\Delta w\ge k^2w\) at all points whose distance to the phase boundary is at least \(R_0\). If the distance of \(x\) to that boundary is \(d>R_0+1\), this inequality holds throughout \(B_{d-R_0}(x)\). Let \[A_k(r)=\frac1{|\mathbb S^{n-1}|} \int_{\mathbb S^{n-1}}e^{kr\theta_1}\,\,\mathrm d\theta.\] The radial function \(A_k(|y-x|)/A_k(d-R_0)\) solves \(\Delta a=k^2a\) and equals one on that ball’s boundary. Comparison gives \(w(x)\le A_k(d-R_0)^{-1}\). A spherical cap of angular radius \(r^{-1/2}\) about \(\theta_1=1\) shows, for \(r\ge1\), \(A_k(r)\ge c_k r^{-(n-1)/2}e^{kr}\). Absorbing the polynomial factor into \(e^{\gamma r/2}\), and using \(q\le2w\), proves the assertion. All balls in this proof may be chosen with closure inside \(\Omega\) and then increased by a limit, so no boundary regularity of \(\Omega\) is needed. ◻

Proof. Lemma 8 gives all normalized derivative bounds on fixed balls and a positive local lower bound for \(D_j/\epsilon_j\). Since \(U_j(0)=0\), \(p_j(0)=\sqrt{1-\epsilon_j}\,e_n\), and \(T_j\to0\) locally smoothly, the first convergence follows as well. The identities \[T_jp_j=-\tfrac12\nabla D_j,\qquad (p_j\cdot\nabla)T_j=-T_j^2-\tfrac12\nabla^2D_j\] give \(\mathcal T e_n=0\) and \(\partial_s\mathcal T=0\) after division by \(\sqrt{\epsilon_j}\). Equation (31) gives \(\mathop{\mathrm{tr}}\mathcal T=0\). Twice differentiating it gives \(\Delta T_j=-\sqrt2\nabla^2(v_jD_j)=O(\epsilon_j)\) on every fixed ball, including every fixed derivative order. Thus \(\Delta'\mathcal T=0\). The limiting \(D\) equation is \[(-\Delta+2\sqrt2 g(s)\partial_s+2G(s))\mathcal D =2|\mathcal T(x')|^2.\] Conjugation by \(G\), using \(G'/G=-\sqrt2 g\) and \(G''=(3g^2-1)G\), gives (44). Finally (42), \(G(0)=1\), and \(G'(0)=0\) give (45). ◻

Proof. There are four steps. We remove the source by monotone exhaustion, classify the extreme positive homogeneous solutions, compute their normal derivative-to-mass ratios, and then pass to all solutions on a single finite interval.

The even inhomogeneous part. For each integer \(R\) solve \(\mathcal L_0 E_R=S\) on \(B_R(0)\) with zero Dirichlet data. The potential \(2-G\) lies in \([1,2]\), so comparison gives \(0\le E_R\le F\). Reflection in \(s\) preserves the operator, domain and source, hence \(E_R\) is even in \(s\). Comparison on nested balls gives \(E_R\le E_{R+1}\) on \(B_R\). Interior estimates and the upper bound by \(F\) show that their increasing limit \(E\) is smooth locally and solves the stated equation. This proves (46). This construction does not interchange an uncontrolled source integral with a limit.

The homogeneous extreme points. It remains to control \(h\) in (46): \(E\) contributes no normal derivative at zero and only increases the weighted mass. The set \(\mathcal K\) is convex and compact in the topology of smooth convergence on compact sets: Harnack and interior estimates bound every derivative on each ball, and the normalization is preserved in a limit. Every nonzero nonnegative homogeneous solution is strictly positive. For a horizontal translation \(t\in\mathbb R^{n-1}\), uniform Harnack chains give \[h(x'+t,s)\le C_t h(x',s)\quad\hbox{for all }(x',s),\] with \(C_t\) independent of the base point. The operator is invariant under these translations. If \(h\) is extreme in \(\mathcal K\), choose \(C>C_t\); both \(C^{-1}h(x'+t,s)\) and \(h(x',s)-C^{-1}h(x'+t,s)\) are nonnegative homogeneous solutions. Normalizing these two summands at the origin writes \(h\) as a convex combination in \(\mathcal K\). Extremality therefore makes the translate proportional to \(h\). The positive proportionality factors form a continuous multiplicative character of \(\mathbb R^{n-1}\), so they equal \(e^{b\cdot t}\) for some real \(b\). Consequently \(h(x',s)=e^{b\cdot x'}f(s/\sqrt2)\) and \[ -f''-2\mathop{\mathrm{sech}}^2 y\,f=-(4-2|b|^2)f. \tag{50}\]

A positive solution \(f\) gives the ground-state identity \[\int_\mathbb R\big(|\zeta'|^2-2\mathop{\mathrm{sech}}^2y\,\zeta^2 +(4-2|b|^2)\zeta^2\big)\,\,\mathrm dy =\int_\mathbb Rf^2\left|\left(\frac\zeta f\right)'\right|^2\,\,\mathrm dy\ge0 \quad(\zeta\in C_c^1(\mathbb R)).\] Use compactly supported approximations to \(\mathop{\mathrm{sech}}y\), for which \((-\partial_y^2-2\mathop{\mathrm{sech}}^2y)\mathop{\mathrm{sech}}y=-\mathop{\mathrm{sech}}y\). It follows that \(4-2|b|^2\ge1\); its upper bound \(4\) is immediate. For \(a=\sqrt{4-2|b|^2}>1\), two linearly independent solutions of (50) are \(f_a^\pm(y)=e^{\pm ay}(a\mp\tanh y)\). They are positive. The coefficient of \(f_a^+\) in any positive solution is nonnegative by its asymptotic behavior at \(+\infty\), and the coefficient of \(f_a^-\) is nonnegative by the behavior at \(-\infty\). Extremality excludes a combination with both coefficients positive. At \(a=1\), reduction of order gives the basis \[\mathop{\mathrm{sech}}y,\qquad \mathop{\mathrm{sech}}y\int_0^y\cosh^2t\,\,\mathrm dt.\] The second member dominates with opposite signs at the two ends, so a positive solution is a multiple of \(\mathop{\mathrm{sech}}y\). This proves the stated necessary classification of extreme points.

The sharp full-line coefficient. We compute the derivative-to-mass ratio for the unnormalized \(f_a^+\); reflection gives the other sign. Define \[M(a)=\frac34\int_\mathbb Re^{ay}\mathop{\mathrm{sech}}^4y\,\,\mathrm dy,\qquad 1\le a\le2.\] Integration by parts, with boundary term zero because \(a<4\), gives \[\frac1\sigma\int_\mathbb RG(s)^2f_a^+(s/\sqrt2)\,\,\mathrm ds =\frac{3a}{4}M(a),\qquad \left|\frac{\,\mathrm d}{\,\mathrm ds}f_a^+(s/\sqrt2)\Big|_{s=0}\right| =\frac{a^2-1}{\sqrt2}.\] Thus the ratio of half the derivative to this mass is \[ \frac{\sqrt2}{3}\frac{a-1/a}{M(a)}. \tag{51}\] The function \(M\) is convex. Its value and derivative at \(2\) can be computed without special-function formulas: with \(t=(1+\tanh y)/2\), \[M(2)=6\int_0^1t^2\,\,\mathrm dt=2,\qquad M'(2)=3\int_0^1t^2\log\frac{t}{1-t}\,\,\mathrm dt=\frac32.\] For the last equality the two logarithmic integrals are \(-1/9\) and \(-11/18\), respectively. The tangent-line inequality for convex \(M\) therefore yields \[ M(a)\ge\frac32a-1\ge\frac43(a-1/a),\qquad 1\le a\le2, \tag{52}\] because the difference in the second inequality is \((a-2)(a-4)/(6a)\ge0\). The ratio in (51) is at most \(1/(2\sqrt2)\).

One finite interval for the entire cone of positive solutions. The full-line mass need not be continuous in the local smooth topology. We therefore pass to a finite interval before applying convex compactness. The normalized functions \(f_a^\pm/a\), including \(a=1\), form a compact family, and \[\mathop{\mathrm{sech}}^4y\,\frac{f_a^\pm(y)}{a}\le C e^{-2|y|} \quad(1\le a\le2).\] Their full weighted masses have a strictly positive uniform lower bound. Consequently, for every \(c>1/(2\sqrt2)\), a single sufficiently large \(P\) makes the two linear inequalities \[\pm\frac12h_s(0)\le\frac c\sigma \int_{-P}^P G(s)^2h(0,s)\,\,\mathrm ds\] true for every extreme point of \(\mathcal K\). Both sides are continuous linear functionals in the local smooth topology. The Krein–Milman theorem extends these inequalities from the extreme points to their closed convex hull, which is \(\mathcal K\). Scaling also covers every nonnegative homogeneous solution, including zero. Since \(E_s(0)=0\) and \(E\ge0\), the same inequality holds for \(F=E+h\). This proves (48). It implies (47) by bounding the truncated mass by the full mass and sending \(c\downarrow1/(2\sqrt2)\). ◻

Proof. Choose \(c_1\) strictly between \(1/(2\sqrt2)\) and \(c\), and fix a finite \(P\ge1\) supplied by Lemma 11 for \(c_1\). If no uniform \(\delta\) existed for this \(P\), there would be solutions and zeros with \(0<\epsilon_j=D_j(z_j)\to0\) violating (54). Apply Lemma 10 to translate and rotate them. On the fixed interval \([-P,P]\), smooth convergence gives \[\frac1{\epsilon_j}\int_{-P}^P q_j(se_n)^3D_j(se_n)\,\,\mathrm ds \longrightarrow\int_{-P}^P G(s)^3\mathcal D(0,s)\,\,\mathrm ds =\int_{-P}^P G(s)^2F(0,s)\,\,\mathrm ds>0.\] The source in (44) is nonnegative and even in \(s\), so the limiting mean satisfies the same bound with coefficient \(c_1\). The violated inequalities and (45) instead give the reverse weak inequality with coefficient \(c\). The displayed strictly positive mass and \(c_1<c\) contradict each other. All compactness constants depend only on \(n\) and the chosen finite interval, so the resulting threshold is uniform over the entire class of solutions. Finally the integrand is nonnegative, which proves the assertion about larger intervals. ◻

Proof. We prove the assertions in their order: first tube injectivity and defect mass, then the radial tail, then the surface mass retained outside the bad set, and finally the Hardy coefficient.

Tube injectivity and defect mass. Curvature \(O(1/s)\) and profile convergence give the local tube description. To verify injectivity, a collision would have base points within distance \(2P\), hence in a common good chart when \(S\) is large. Different sheets cannot collide because their separation tends to infinity; on a single sheet the small slope and curvature give uniqueness of the normal projection on this fixed-width tube. The volume Jacobian is \(\det(\mathrm{Id}+tA)=1+o_S(1)\). A unit tube, on which \(q\) is bounded below, controls surface area by energy. Bounded density therefore gives the area bound for arbitrary centers. Local Harnack comparison in (36) also gives the first inequality of (57). To obtain its second inequality, apply (35) with a cutoff equal to one on \(B_{r+1}(x)\), supported in \(B_{2r+2}(x)\), and with gradient bounded by \(C/r\); use \(q^2\le4e(v)\). On good zeros the identity \(H=\partial_\nu D/(2|p|^2)\) gives \(|H|\le CD\).

The radial error has a vanishing tail. Integrating the second term in (6) gives \[\int_{\{|x-y|>R_0\}}|\partial_rv|^2|x-y|^{-m}\,\,\mathrm dx \le s_0\omega_m\bigl(K-M_{R_0}(y)\bigr).\] In a unit tube, \(|p(\Phi(z,t))-\nu(z)|\le C\sqrt{D(z)}\), by (36); changing radial directions costs \(C/r\). After squaring, integrating in \(t\), and using the unit tube’s positive lower profile bound, the surface radial-normal integral is bounded by this ambient flux tail, plus \[C\int_{\Sigma_S\cap\{r>R_0\}} \bigl(Dr^{-m}+r^{-m-2}\bigr)\,\,\mathrm dS.\] On a dyadic shell of radius \(a\), (57) bounds the first error by \(Ca^{-2}\), and the area bound does the same for the second. The mean-curvature term in (58) is at most \(Ca^{-1}\) on that shell. Summing proves the explicit uniform bound \[\eta(R_0)\le C\left( \sup_{y\in Z}\bigl(K-M_{R_0-2}(y)\bigr)+R_0^{-1}\right),\] which tends to zero by uniform density convergence at \(Z\). The constants can be fixed when the unit tube and (57) are first valid, at \(S=S_*\). Increasing \(S\) merely restricts the nonnegative integrals to smaller subsets, so it does not worsen this bound.

Removing the bad set while retaining the coefficient \(K\). We first construct the stated smooth cutoff without differentiating the nonsmooth distance \(s\). Mollify \(s\) at radius \(S/8\), writing \(\widetilde s=\rho_{S/8}*s\), and choose a smooth nondecreasing \(\theta:\mathbb R\to[0,1]\) equal to zero on \(( -\infty,5/4]\) and to one on \([7/4,\infty)\). Then \(\chi=\theta(\widetilde s/S)\) has the required properties: \(|\widetilde s-s|\le S/8\), so its derivatives are supported in \(9S/8\le s\le15S/8\), strictly inside the regular region \(s>S\). Moreover \(|\nabla\chi|\le C/S\) and \(|\nabla^2\chi|\le C/S^2\).

The same tube and the fixed-thickness bad-set estimate imply \[ \sup_{y\in Z}\mathcal H^m\bigl(\Sigma\cap\{S<s<2S\}\cap B_r(y)\bigr) =o(r^{m-2}) \quad(r\to\infty) \tag{61}\] for each fixed sufficiently large \(S\). Indeed the unit tube over this set lies in \(B_{2S+1}(Z)\cap B_{r+1}(y)\).

To prove (59), first observe the exact identity \[e(v)=\frac{q^2}{2}-\frac{q^2D}{4}.\] Together with Lemma 14 this gives, uniformly at bad centers, \[ R^{-m}\int_{B_R(y)}q^2\longrightarrow \sigma K\omega_m. \tag{62}\] Its radial distributional consequence is \[R^{-m}\int q^2F(|x-y|/R)\,\,\mathrm dx \longrightarrow \sigma K m\omega_m\int t^{m-1}F(t)\,\,\mathrm dt.\] For completeness, the part of this integral at distance greater than \(P\) from the zero set is \(o_P(1)R^m\), uniformly for \(R\ge1\). Take a maximal unit-separated net on \(\Sigma\). A fixed small ball about any zero has energy bounded below: the uniform gradient bound keeps \(|v|\le1/2\) in a uniform neighborhood of that zero. Disjoint such balls and the density bound give at most \(C(a+1)^m\) net points in any ball of radius \(a\). The portion with zero-set distance in \([j,j+1]\), inside \(B_{2R}(y)\), is therefore covered with volume at most \[C(R+j+2)^m(j+2)^n.\] Exponential decay (41), summed over \(j\ge P-1\), proves the assertion.

For fixed \(P\), points within \(P\) of a zero with \(s\le2S\) have volume \(o(R^{m-2})\) in \(B_{2R}(y)\), by the fixed enlargement of the bad set. Every remaining point within \(P\) of \(\Sigma\) has a nearest zero in \(\{\chi=1\}\), and belongs to the injective good tube. Replacing \(F(|\Phi(z,t)-y|/R)\) by \(F(|z-y|/R)\) costs \(O(P/R)\) in normalized mass. The normal profile integral is at most \(\sigma+o_S(1)\). First choose \(P\) to make the discarded tail small, then \(S\) to make this profile and Jacobian error small. To make the direction of this transfer explicit, the preceding estimates give \[R^{-m}\int q^2F(|x-y|/R)\,\,\mathrm dx \le (\sigma+o_S(1))R^{-m}\int_\Sigma\chi F(r/R)\,\,\mathrm dS +o_P(1)+o_{R\to\infty;P,S}(1).\] The error \(o_P(1)\) tends to zero as \(P\to\infty\); with \(P\) fixed, \(o_S(1)\) tends to zero as \(S\to\infty\). The radial mass limit therefore gives (59) in precisely the stated order of choices.

The Hardy coefficient. The radial \(q^2\) mass has now been identified, including its coefficient. Put \(a=(m-2)/2\). In (35) use \(\eta(x)=|x-y|^{-a}\) on \(R_0/2<|x-y|<4R\), extended with smooth cutoffs on the adjacent inner and outer dyadic shells. The cutoff-shell costs are \(O(1)\), uniformly in \(y\) and \(R\); this follows from \(\int_{B_t(y)}q^2\le Ct^m\). The remaining right-hand integral is \[a^2\int_{\{R_0/2<|x-y|<4R\}}q^2|x-y|^{-m}\,\,\mathrm dx.\] Stieltjes integration by parts in (62) gives \[\int_{\{b<|x-y|<cR\}}q^2|x-y|^{-m}\,\,\mathrm dx =\sigma K m\omega_m\log R+o(\log R)\] for fixed \(b,c>0\). The same leading term holds after including the cutoff shells. Division by \(\sigma\log R\) proves (60). ◻

Proof. Defining the adjacent foot. All \(K\ge2\) graphs have height \(o(s)\); their separation diverges, so the nearest distinct-sheet distance has the claimed size. Orientations alternate because the solution changes sign on crossing each regular zero graph. Equivalently, \(l\) is the nearest distance to a zero whose oriented normal has scalar product less than \(-1/2\) with \(\nu(z)\). The minimum is attained within \(o(s)\), and every competitor that could improve it lies in the same buffered chart. A nonadjacent sheet cannot minimize: the chord to it crosses an intervening zero sheet at a strictly smaller distance. This also proves that the definition agrees on overlapping charts.

Projection to an adjacent sheet is smooth at these distances. Indeed its curvature is \(O(s^{-1})\), the distance is \(o(s)\), and the slopes are small. In graph coordinates the squared-distance function has positive definite Hessian near its minimizing foot; all possible minimizers lie in this region, since their horizontal displacements are at most \(l=o(s)\). It thus has a unique foot.

The branch Laplacian. Choose on the target graph the unit normal \(N\) pointing toward the source. Then \(z=z'+l_bN(z')\), and the distance Hessian at \(z\) has eigenvalues \[\frac{\kappa_i}{1+l_b\kappa_i}\le\kappa_i\] in the transported target tangent directions, and zero in the normal direction. Here \(1+l_b\kappa_i>0\); its operator norm is \(O(s^{-1})\). The inequality follows exactly by subtracting \(l_b\kappa_i^2/(1+l_b\kappa_i)\). Restricting to the source surface removes its normal-normal Hessian component. Its absolute value is bounded by \[\frac Cs\bigl(1-(\nu(z)\cdot N)^2\bigr) =\frac Cs|\nabla_\Sigma l_b|^2.\] The restriction formula for the Laplacian adds \(-H(z)\nu(z)\cdot N\), of size at most \(|H(z)|\). Taking the ambient trace and bounding the target trace by \(|H(z')|\) proves (63).

The Jacobian and the count of two. The exact projection area Jacobian is \[ J_{\pi_b}(z) =\frac{|\nu(z)\cdot N(z')|} {|\det(\mathrm{Id}+l_b A_N(z'))|} =1+o_S(1). \tag{64}\] To check multiplicity globally, fix a target point \(z'\). Write \(l(z)\le\varepsilon_S s(z)\) on \(\Sigma_S\), where \(\varepsilon_S\to0\). The Lipschitz property of \(s\) gives \[|z-z'|\le\frac{\varepsilon_S}{1-\varepsilon_S}s(z').\] Choose \(S\) so this fraction is smaller than the fixed inner radius of a good chart. Every active source projecting to \(z'\) then lies in that single chart centered at \(z'\). The fixed normal line through \(z'\) meets each small-slope graph at most once. Any intervening sheet would meet the source-target segment and contradict minimality. Only the immediately adjacent sheet on either side can therefore supply a source, giving at most two preimages. This reasoning also covers ties between active branches. A countable atlas and a measurable selection among the finite local branches permit the area formula with this multiplicity. Finally \(s\) and distance to a fixed \(y\in Z\) are 1-Lipschitz and \(s(z)\le|z-y|\); the two ratio estimates follow from \(l=o(s)\). ◻

Proof. The cap can affect the radial geometric error, but it must not multiply the errors near the fixed inner boundary or the bad set. We keep those three terms separate. Choose a smooth radial \(\alpha\), zero on \(r\le R_0\), one on \(r\ge2R_0\), with derivatives bounded by \(C R_0^{-j}\). Test (65) against the nonnegative compactly supported \(\alpha\chi\phi_R\). Its support lies in the regular open surface, so distributional integration by parts has no singular-set boundary. For a radial function \(f\), the convention \(H=\operatorname{div}_\Sigma\nu\) gives \[\Delta_\Sigma f =f''+(m-1)f'/r +(f'/r-f'')(e_r\cdot\nu)^2-Hf'(e_r\cdot\nu).\] Thus the error between \(\Delta_\Sigma\phi_R\) and its flat \(m\)-dimensional radial Laplacian is at most \[C\bigl((e_r\cdot\nu)^2r^{-m}+|H|r^{1-m}\bigr).\] It contributes at most \(T_0\eta(R_0)\), by (58).

Terms differentiating \(\alpha\) are supported where \(r\le2R_0\), and there \(s\le r\). The uniform bound \(l=o(s)\), after fixing its lower threshold \(S\), implies \(w\le\psi(l)\le C_{R_0,S}\) on this set, independently of the cap. Area and derivative bounds there give a cap-independent constant error. Terms differentiating \(\chi\) are supported in \(S<s<2S\), where again \(w\le C_S\), independently of the cap. The ambient derivative bounds for \(\chi\) and the bounded mean curvature imply that these terms are bounded in absolute value by \[C_S\int_{\Sigma\cap\{S<s<2S,\ R_0<r<2R\}} r^{2-m}\,\,\mathrm dS.\] On a dyadic shell of radius \(a\), (61) bounds this by a quantity tending to zero as \(a\to\infty\), uniformly in \(y\). The sum of \(O(\log R)\) such quantities is \(o(\log R)\): split at any fixed large shell, then use the uniform small bound on all later shells. This is the required Cesaro estimate.

On the support of \(F(r/R)\), \(\alpha=1\) for large \(R\). The main term on the left is therefore exactly that in (66). Bound the nonnegative right-hand test by \(r^{2-m}\); all remaining terms have just been estimated. ◻

Proof. Use \(w=\min\{T_0,\sqrt l\}\). The Green inequality will bound its average by \(O(\log R)\). A Harnack estimate for adjacent height differences then turns this average bound into a bound at any chosen good zero.

The average estimate. The curvature integral in (66) is at most \(C(1+\log R)\). Indeed \(|H|\le CD\), and each dyadic shell contributes \(O(1)\) by (57). For its foot term use (64), multiplicity two and the comparable radial weights; the image is contained in \(\Sigma_{S/2}\cap\{R_0/2<r<4R\}\). The same shell estimate applies.

From one gap to a patch of comparable gaps. Given a good zero \(z\), choose a nearest \(y\in Z\), and set \(R=s(z)=|z-y|\), \(T_0=\sqrt{l(z)}\). We claim that either \(l(z)\le C'R^{1/10}\), or an area-\(cR^m\) patch of the source sheet, within distance \(cR\) of \(z\), has \(l\ge c l(z)\). In the chart of Lemma 14, an adjacent height difference \(h=f_{i+1}-f_i>0\) satisfies \[\operatorname{div}(a(x')\nabla h)=\mathcal M(f_{i+1})-\mathcal M(f_i),\qquad \mathcal M(f)=\operatorname{div}\frac{\nabla f}{\sqrt{1+|\nabla f|^2}},\] where \[a(x')=\int_0^1 D_\xi\!\left(\frac{\xi}{\sqrt{1+|\xi|^2}}\right) \bigl(\nabla f_i+t(\nabla f_{i+1}-\nabla f_i)\bigr)\,\,\mathrm dt.\] The small slopes make this matrix uniformly elliptic. The right-hand side has absolute value at most \(CR^{-2+1/10}\). Interior Harnack for a positive solution with bounded forcing, after rescaling to a unit disk, therefore gives on a fixed smaller disk \[\inf h\ge c h(x'_z)-C R^{1/10}.\] Every adjacent vertical gap relevant to the source sheet is at least \(l(z)\) at \(x'_z\). If \(C'\) is sufficiently large, the asserted lower bound follows for both adjacent gaps, or for the sole adjacent gap on an extreme sheet. Vertical gaps and nearest Euclidean distances are comparable uniformly by the small slope bound; potential closest points outside the buffered chart are at distance comparable to \(R\), whereas \(l=o(R)\). This proves the claim.

Shrink the patch fraction so \(F(r/R)\ge c>0\) on it. On this patch \(s\ge(1-c)R\), hence \(\chi=1\) for large \(R\), and \(w\ge cT_0\). Thus the left side of (66) is at least \(c'T_0\). Fix \(R_0\) large enough that \(\eta(R_0)<c'/2\), independently of \(z,T_0,R\). Absorb its last term and obtain \(T_0\le C(1+\log R)\). Squaring proves (67) in the second case, and its first case already has the asserted bound. ◻

Completion of the proof of Theorem 3. The nearly linear gap test. Fix a small tolerance \(\delta>0\) and use \[\psi(l)=l+\log l,\qquad w=\min\{T_0,\psi(l)\}.\] Because \(l\to\infty\), \(\psi'\le1+\delta\) once \(S\) is large. Moreover \[-\frac{\psi''(l)}{\psi'(l)}=\frac1{l(l+1)},\qquad \frac{l(l+1)}{s}\longrightarrow0\] by (67). Thus (65) and Lemma 17 apply with \(L\le1+\delta\).

The upper logarithmic coefficient. Let \(1/(2\sqrt2)<c_*<\sqrt2/3\) and \(P\) be those of Lemma 13. Increase \(S\) so this lemma applies on \(\Sigma_{S/2}\) and its \(P\)-tube is injective. The source term of (66) contributes one curvature integral; the target terms contribute at most two by Lemma 16. The area Jacobians and radial-weight changes cost a relative \(1+C\delta\) after increasing \(S\). Thus the curvature integral in (66), including its factor \(L\), is at most \[(1+C\delta)\,3\int_{\Sigma_{S/2}\cap\{3R_0/4<r<3R\}} |H(z)|r^{2-m}\,\,\mathrm dS.\] At every such base point, \[|H(z)|\le\frac{c_*}{\sigma} \int_{-P}^{P}q^3D(z+t\nu(z))\,\,\mathrm dt.\] Choose \(R_0\) also large compared with \(P\). Tube changes of radial weights and volume Jacobians have relative error as small as desired, and this enlarged region’s \(P\)-tube lies in \(R_0/2<|x-y|<4R\) when \(R_0>4P\). Using (60) therefore gives \[ \limsup_{R\to\infty}\frac{L}{\log R} \int_{\Sigma_S\cap\{R_0<r<2R\}} r^{2-m}\bigl(|H(z)|+|H(\pi(z))|\bigr)\,\,\mathrm dS \le (1+C\delta)\,3c_*K m\omega_m\frac{(m-2)^2}{4}. \tag{68}\] There is no extra multiplicity from this last transfer: the normal tube over \(\Sigma_{S/2}\) is injective.

Almost all surface mass reaches the cap. For the lower bound choose \(T_0=(\sqrt2-\delta)\log R\). We first prove that, for any \(\gamma>0\), \[ D(z)+D(\pi(z))\ge c_\gamma e^{-(\sqrt2+\gamma)l(z)} \quad(z\in\Sigma_S) \tag{69}\] when \(S\) is large. The endpoint normals have opposite orientations and \(D=o(1)\), so \(|p(z)-p(\pi(z))|\ge c>0\). Apply (36) with exponential rate \(2\sqrt2+2\gamma\) along each half of the chord. Since \(|\nabla p|=|T|\le C\sqrt D\), integration from the two endpoints to the midpoint gives \[c\le C_\gamma \bigl(\sqrt{D(z)}+\sqrt{D(\pi(z))}\bigr) e^{(\sqrt2+\gamma)l(z)/2}.\] Squaring proves (69).

Choose \(\gamma>0\) so \(\beta=(\sqrt2+\gamma)(\sqrt2-\delta)<2\). On the exceptional set \[E_R=\{z\in\Sigma_S:R/2<r<2R,\ l(z)<T_0\},\] (69) gives \(D(z)+D(\pi(z))\ge cR^{-\beta}\). By (57), the area formula, multiplicity two and the radial comparability of the feet, \[\mathcal H^m(E_R) \le CR^\beta\int_{E_R}\bigl(D(z)+D(\pi(z))\bigr)\,\,\mathrm dS \le CR^{m-2+\beta}=o(R^m).\] Outside this exceptional set on the support of \(F\), \(l\ge T_0\); since \(l>1\), \(w=T_0\). Applying (59), with its error chosen at most \(\delta\), gives \[\liminf_{R\to\infty}\frac{R^{-m}}{\log R} \int_\Sigma w\chi F(r/R)\,\,\mathrm dS \ge(\sqrt2-\delta)\bigl(Km\omega_m(m-2)-\delta\bigr).\] The incompatible coefficients. All cutoff errors in (66) other than \(T_0\eta(R_0)\) vanish after division by \(\log R\). To make the order of choices explicit, first fix \(\delta\); then fix \(P,S\) to meet the mean-curvature, tube, concavity and mass requirements; next fix \(R_0\), as large as needed for the radial tail; finally let \(R\to\infty\). Letting \(R_0\to\infty\) in the resulting numerical inequality removes \((\sqrt2-\delta)\eta(R_0)\). Now send \(\delta\to0\) and use (68). Cancellation of the positive factor \(K m\omega_m(m-2)\) yields \[\sqrt2\le \frac{3c_*}{4}(m-2).\] This is impossible for \(3\le m\le6\), since its right side is at most \(3c_*<\sqrt2\). In the endpoint \(m=6\), the strict inequality in Lemma 13 is essential. No critical solution exists, proving Theorem 3. ◻

Proof. Theorem 3 makes every finite \(K_f\) subcritical. Apply (Florit-Simon and Serra 2025, Theorem 3.2) with its tolerance \(\delta(K_f,n)\) and \(r^{-1}\le\delta\); its allowed density is \(K_f+\delta\), so the stated bound suffices. Its sheeting assumptions and (Florit-Simon and Serra 2025, Theorem 2.14, Theorem 4.8, and Lemmas 4.9–4.10) give the remaining assertions, as specified above. Constants \(\pm1\) have an empty core band and cause no exception. ◻

Proof. Local smooth graph representation bounds the quotients \(|\nu(z)\cdot(y-z)|/|y-z|^2\) for nearby zeros; competitors outside a fixed small ball have a positive lower bound because \(\Gamma\ge1\). This proves positivity. If every zero lay in the tangent plane at \(z\), phase decay (41) would bound the energy by \(C e^{-c\mathop{\mathrm{dist}}(x,T_z\Sigma)}\), giving finite density. Thus a finite competitor exists. A signed unit geodesic tangent to a principal direction gives \(\ell/s\to2/|\lambda_i|\) whenever \(\lambda_i\ne0\), proving the first inequality. The second uses the upper penalty bound: \(d\le\ell\Gamma(s)\le C_\Gamma\ell/s\).

If \(D(z_j)\to0\), the translated, rotated fields \(U\) converge smoothly on every fixed ball to \(x_n\), by (36) and the equation. In \(B_{4J}\) all zeros are therefore one graph whose curvature and slope tend to zero. A competitor of value at most \(J\) has \(\ell\le J\); on this graph \(s\le o(1)\ell\), whereas \(\ell\Gamma(s)\ge\ell/s\). This is impossible, also for competitors converging to the diagonal. The contradiction formulation proves uniformity of the threshold. Finally, the second inequality in (72) excludes zeros from each ball \(B_{c_\Gamma d}(z\pm c_\Gamma d\nu)\) when \(c_\Gamma\) is sufficiently small: a point in either ball has \(|\nu\cdot(y-z)|>|y-z|^2/(2c_\Gamma d)\). ◻

Proof. Slab confinement and one-step density control. Translate \(z\) to \(0\) and take \(\nu(z)=e_n\). By (72), the zeros in \(B_{3R}\) lie in \(|x_n|\le C R^2/d\). With \(\mathbb S=e(v)\mathrm{Id}-\nabla v\otimes\nabla v\), the stress tensor identity \(\operatorname{div}\mathbb S=0\), tested against \(x_ne_n\zeta^2\), gives \[\int\zeta^2\mathbb S_{nn} =-2\int x_n\zeta\,\mathbb S e_n\cdot\nabla\zeta.\] The identities \(\nabla v=qp/\sqrt2\) and \(e(v)=q^2(1+|p|^2)/4\) show \(\mathbb S_{nn}\ge c q^2(1-p_n^2)\) and \(|\mathbb S e_n|^2\le Cq^2\mathbb S_{nn}\). Young’s inequality, with a cutoff equal to one on \(B_R\) and supported on \(B_{2R}\), consequently proves \[\int_{B_R}q^2(1-p_n^2) \le C\int_{B_{2R}}q^2(x_n/R)^2.\] On the part of \(B_{2R}\) with \(|x_n|>CR^2/d+\sqrt R\), every zero either lies outside \(B_{3R}\) or has the confined height, so its distance is at least \(c\sqrt R\). Phase decay controls the energy there by \(C R^n e^{-c\sqrt R}\). Elsewhere \((x_n/R)^2\le C((R/d)^2+R^{-1})\).

In the monotonicity formula between \(R/B\) and \(R\), with a fixed \(B\ge8\), the discrepancy is bounded by \(Cq^2(1-p_n^2)\), and the squared radial derivative on these annuli by \(C_Bq^2(1-p_n^2+(x_n/R)^2)\). The preceding estimate, at a fixed multiple of \(R\), therefore gives \[ 0\le M_R(z)-M_{R/B}(z) \le C\bigl((R/d)^2+R^{-1}\bigr)M_{BR}(z) \qquad(10\le R\le c d). \tag{74}\] The exponentially small remainder was absorbed using \(M_R(z)\ge c\) for \(R\ge1\). This uniform lower bound follows from \(v(z)=0\), the uniform first derivative bound, and monotonicity.

Iterating with the larger-radius term. We now retain the delay in (74); replacing \(M_{BR}\) by \(M_R\) would give an unjustified recursion. Set \(R_j=dB^{-j}\), \(k_j=M_{R_j}(z)\), and \(b_j=C(B^{-2j}+R_j^{-1})\). Beyond a fixed starting index, \[k_j-k_{j+1}\le b_jk_{j-1}.\] Choose \(R_{\rm fix}\) and that starting index so \(b_j<1/8\) while \(R_j\ge R_{\rm fix}\). Put \(J=\lceil C_0\sqrt{\log(K(d)+1)}\rceil\). For sufficiently large \(d\), \(R_{2J+10}\ge d^{1/2}\), because \(K(d)\le1+Cd\). Suppose a halving \(k_{j+1}<k_j/2\) occurs at some \(j\ge J-2\) before the fixed-scale endpoint. Then \(k_{j-1}>k_j/(2b_j)>4k_j\), so a halving also occurs at the preceding index. Induction propagates this backwards through all indices in \([\lceil J/2\rceil,J-3]\). On this interval \(b_i\le C e^{-cJ}\): the \(B^{-2i}\) term has this bound, and \(R_i^{-1}\le d^{-1/2}\) is smaller. Multiplying the preceding lower growth factors over at least \(cJ\) indices gives \(K(d)\ge c\exp(c'J^2)\), contradicting the choice of large \(C_0\). Here all \(k_j\) before the endpoint are bounded below by the same positive constant. Thus no such halving occurs.

It follows that \(k_{j-1}\le2k_j\) for \(j\ge J\) up to that endpoint, and hence \(k_{j+1}\ge(1-2b_j)k_j\). The sum of \(b_j\) on this range is uniformly bounded: the first terms form a geometric series, and the \(R_j^{-1}\) terms form a reversed geometric series ending at a fixed radius. The product of \((1-2b_j)^{-1}\) is therefore bounded. At the last index \(M_{R_j}\) has a uniform bound from the bounded energy integrand. Consequently \(k_J\le K_0\) with fixed \(K_0\). Taking \(C\) large enough in (73) makes \(h\le R_J\); monotonicity completes the proof. ◻

Proof. Write \(v_{a_0}=g'b\) and use \((-\partial_t^2+V_g)g'=0\). Then \[(-\partial_t^2+V_g)(g'b) =-\frac{1}{g'}\bigl((g')^2b'\bigr)'=1,\] since \((g')^2b'=\int_t^\infty g'(r)\,\,\mathrm dr\). Both terms in the bracket in (75) are nonnegative, and \(g'>0\). Moreover \(g'(t)\sim2\sqrt2e^{-\sqrt2t}\) and \(\int_t^\infty g'=1-g(t)\sim2e^{-\sqrt2t}\), so \(v_{a_0}(t)\to1/2\) as \(t\to\infty\). Continuity gives the positive lower and finite upper bounds, uniformly for \(a_0\) in a compact interval. Differentiating the displayed formula gives a bounded first derivative; the equation and its successive derivatives give all remaining fixed derivative bounds. ◻

Proof. One graph and preliminary bounds. The fixed bound of Lemma 22 is subcritical by Theorem 3. Apply Corollary 19, including its sheeting conclusions, at radius \(h\). Thus the density used here is the fixed local bound \(K_0\); the contradiction hypothesis still allows unbounded density on larger scales. The confinement thickness divided by this radius is \(O(h/d)\). Increasing the fixed constant in (73) makes this ratio as small as required, without changing the density bound. The resulting graphs have arbitrarily small fixed slope, the first estimate in (76), and \[ |H|\le C_\iota h^{-2+\iota}\quad\text{for every fixed }\iota>0. \tag{77}\] An additional graph in an inner cylinder would cross the normal line at \(z\) at distance \(O(h)\), producing a competitor of value \(O(h)\), contrary to \(d/h\) being sufficiently large. Thus just one sheet remains in a smaller cylinder. On fixed windows centered on inner zeros the solution, in oriented normal coordinates, converges uniformly and smoothly to \(g\). The lower gradient bound from sheeting and interior elliptic estimates also give unit-scale bounds on all fixed orders of graph derivatives. These statements are uniform in the centers and solutions, by the stated uniformity of the subcritical ball input and the usual sequential formulation of compactness.

It remains to strengthen the mean-curvature derivatives and then recover the large-scale bound on \(\nabla_\Sigma A\). The two projections below serve different purposes: the first removes the leading odd forcing from the equation at the zero sheet, and the second uses the resulting tangential gain to improve the derivatives of \(H\). We prove the derivative improvement; in particular we do not differentiate (77) at its original scale. Fix an inner zero as origin. Let \(X(y)\) be its tangent-plane graph and use Fermi coordinates \(X(y)+t\nu(y)\). Choose a small number \(\epsilon>0\), to be fixed at the end, and write \(R_h=h^\epsilon\). All arguments below take place in a fixed finite sequence of nested boxes with side lengths fixed fractions of \(R_h\), starting inside \(\{|y|_\infty,|t|<R_h\}\). The much larger \(h\)-scale chart guarantees that these coordinates are injective there, have polynomially bounded unit-scale derivatives of each fixed order, and contain no other zero. The distance to \(\Sigma\) is comparable to \(|t|\) on the boxes. Consequently \[\varphi(y,t)=v(X(y)+t\nu(y))-g(t)=o(1)\] uniformly. To see uniformity also at unbounded \(t\), first choose a large fixed depth and use phase decay beyond it, then use the fixed window convergence at shallower depths. For \(|t|\) a fixed fraction of \(R_h\), \(\varphi\) and every needed fixed derivative are bounded by \(C h^C e^{-cR_h}\): apply phase decay and unit-scale elliptic estimates to \(v\mp1\), and differentiate the coordinate composition.

We record the interpolation used below. If a tensor has uniformly bounded \(C^N\) norm on unit balls and \(C^{1/2}\) seminorm at most \(Ch^{-3/2}\), its derivatives of orders \(1\le k\le30\) are bounded by \(C h^{-5/4}\), by taking \(N\) sufficiently large once and for all. Indeed the standard interpolation exponent of the low seminorm can be taken arbitrarily close to one for any fixed \(k\) by increasing \(N\); subtracting its value at the center removes the constant term. Applied to the coordinate components of \(A\), whose change-of-frame terms have the same or better bound, this gives \[ |\partial_y^k A|\le C h^{-5/4}\quad(1\le k\le30),\qquad |\partial_y^kH|\le C_\epsilon h^{-2+\epsilon}\quad(0\le k\le30). \tag{78}\] For the second assertion use (77) with \(\iota<\epsilon/2\) and interpolate its supremum norm with a sufficiently high unit-scale derivative bound. All interpolation orders are finite and fixed before \(h\) tends to infinity.

In these coordinates the equation is exactly \[ \begin{split} (-\Delta_y-\partial_t^2+V_g)\varphi={}&K_tg'+K_t\varphi_t +(g_t^{ij}-\delta^{ij})\partial_{ij}\varphi +b_t^i\partial_i\varphi-3g\varphi^2-\varphi^3,\\ V_g={}&3g^2-1,\qquad K_t=\mathop{\mathrm{tr}}[A(\mathrm{Id}+tA)^{-1}]\,. \end{split} \tag{79}\] Here \(g_t^{ij}\) is the inverse tangential metric, and \(b_t^i\) is the first-order coefficient in its Laplace operator. Neither superscript nor subscript \(t\) denotes differentiation in this display. Matrix inversion in \(\mathrm{Id}+tA\), the graph metric, and (78) give, with derivatives through order 20, \[ \begin{gathered} K_t=H-t|A|^2+t^2\mathop{\mathrm{tr}}A^3+O(h^{-4+C\epsilon}),\qquad K_t=O(h^{-2+C\epsilon}),\\ g_t^{ij}-\delta^{ij}=O(h^{-1+C\epsilon}),\qquad b_t^i=O(h^{-1+C\epsilon}). \end{gathered} \tag{80}\] Every occurrence of \(C\epsilon\) in this proof has a bounded absolute multiplier independent of \(\epsilon\). For example the tangent slope is \(O(h^{-1+\epsilon})\), and the remainder in the first line contains at least four factors of \(A\) and at most a fixed power of \(|t|\). The differentiated remainders obey the same bounds, since each positive derivative of \(A\) is no larger than \(O(h^{-1})\).

A first bound for the profile error. Use \(v_0\) from Lemma 23 on \(t\ge0\) and its reflection on \(t\le0\), separately on the two half-boxes. The reflected pair need not be differentiable at zero; no full-line supersolution is asserted. In each half-box move the metric and drift terms to the left of (79) and write the nonlinearity as \((3g\varphi+\varphi^2)\varphi\). Its coefficient tends uniformly to zero, so the resulting operator applied to \(v_0\) is at least \(1/2\). Comparison is valid after division by this positive supersolution, which makes the zeroth-order coefficient positive. For each lateral or distant normal face, \(v_0e^{-c_0\,\text{distance to face}}\) is also a supersolution when \(c_0\) is a sufficiently small fixed number. The sum of these face barriers and \(C h^{-2+C\epsilon}v_0\) dominates both signs of \(\varphi\). The bottom boundary has \(\varphi(y,0)=0\). Shrinking the box makes all face contributions exponentially small. Interior elliptic estimates across \(t=0\), using the smooth equation, then give \[ \|\varphi\|_{C^{15}}\le C_\epsilon h^{-2+C\epsilon}. \tag{81}\] The coefficient norms needed for this bootstrap are uniform by (80); the nonlinearity is smooth and has already been bounded in supremum norm.

The inverse on the complement of the translation mode. The positive barrier controlled the full profile error. The next estimate separates its translation mode from the remaining normal modes, where a spectral gap is available. Let \(L_t=-\partial_t^2+V_g\). It has kernel spanned by \(g'\) and a fixed spectral gap \(\lambda>0\) on its \(L^2(\mathbb R)\) orthogonal complement. One elementary justification is the ground-state factorization of its quadratic form, simplicity of the positive zero mode, and compactness on bounded intervals together with \(V_g\to2\) at infinity; a Rayleigh sequence tending to zero could neither escape to infinity nor stay orthogonal to the kernel. Let \(P_\perp\) be this projection. For \[(-\Delta_y+L_t)w=P_\perp F,\qquad P_\perp w=w,\] the regularized \(L_t^2\) norm of each \(y\)-derivative of \(w\) is a subsolution for \(-\Delta_y+\lambda\), with source the corresponding norm of \(F\). This follows by differentiating the squared norm, using \(\langle L_tw,w\rangle\ge\lambda\|w\|_2^2\) and Cauchy–Schwarz, and then letting the regularization vanish. Comparison with the constant source bound plus decaying face barriers controls these norms on each smaller \(y\)-box, with an exponentially small multiple of their boundary bound. Commuting two additional \(y\)-derivatives and using the equation gives the same bound in \(H_t^2\), since \(\|w\|_{H^2}\le C(\|L_tw\|_2+\|w\|_2)\). Thus source derivatives through order \(j+2\) control the value at \(t=0\) and its \(y\)-derivatives through order \(j\). This assertion applies separately to even and odd functions of \(t\), because both the operator and \(P_\perp\) preserve parity. It applies also to a range of strictly positive derivative orders: no estimate of the undifferentiated source is needed to estimate a differentiated equation. Polynomial boundary bounds suffice in every use here.

The whole-line equation and the first parity projection. Choose an even cutoff \(\chi_h(t)\), equal to one on an inner normal interval and supported in a slightly larger interval inside the box, with derivatives supported where \(|t|\) is comparable to \(R_h\). Put \(\Phi=\chi_h\varphi\). If \(E_{\rm raw}\) denotes the right side of (79) minus \(Hg'\), the exact equation on the whole normal line is \[(-\Delta_y+L_t)\Phi=Hg'+E_{\rm full},\qquad E_{\rm full}=\chi_h E_{\rm raw}+(\chi_h-1)Hg' -2\chi_h'\varphi_t-\chi_h''\varphi.\] The kernel tail \((\chi_h-1)Hg'\) and the two commutators, together with all their needed \(y\)-derivatives, are \(O(h^{-N})\) for every fixed \(N\). For the tail this follows from the preliminary derivatives of \(H\) and the exponential decay of \(g'\); for the commutators it follows from the normal-edge estimates. Restoring the cutoff leading term \(-\chi_h t|A|^2g'\) to the whole-line odd function \(-t|A|^2g'\) costs another error of the same order. We henceforth write \(\varphi\) for \(\Phi\), and \(E\) for \(E_{\rm full}\). Write \[\varphi=u_0(y)g'(t)+\varphi^\perp(y,t),\qquad (-\Delta_y+L_t)\varphi=H(y)g'(t)+E(y,t).\] We normalize kernel coefficients by \(\int(g')^2\). From (80) and (81), in \(L_t^2\) with \(y\)-derivatives through order 11, \[ E=-t|A|^2g'+O(h^{-3+C\epsilon}),\qquad E^{\rm even}=O(h^{-3+C\epsilon}). \tag{82}\] Here \(t^2\mathop{\mathrm{tr}}A^3g'\) has size \(h^{-3+C\epsilon}\), every metric error is a coefficient of size \(h^{-1+C\epsilon}\) times a derivative of \(\varphi\) of size \(h^{-2+C\epsilon}\), and the normal-drift and nonlinear errors have size \(h^{-4+C\epsilon}\).

The inverse estimate for the even orthogonal component, followed by evaluation at \(t=0\), bounds \(\varphi^\perp(y,0)\) and its derivatives through order 9 by \(C h^{-3+C\epsilon}\); the odd component vanishes there. Since \(\varphi(y,0)=0\), the same bound holds for \(u_0\). The kernel equation \[ -\Delta_yu_0=H+E_0 \tag{83}\] therefore bounds \(H\) and its derivatives through order 7 by \(C h^{-3+C\epsilon}\). In addition, positive \(y\)-derivatives of \(-t|A|^2g'\) gain \(h^{-1/4}\) over its \(h^{-2}\) bound by (78); derivatives of the remainder in (82) are already smaller. Applying the inverse estimate to the entire orthogonal component consequently gives \[ \|\partial_y^k\varphi\|_{L_t^2} \le C h^{-2-1/4+C\epsilon},\qquad1\le k\le8. \tag{84}\] For these \(L_t^2\) bounds no loss of two derivative orders is required; the already controlled derivatives of \(u_0\) cover the kernel term.

The first projection has now bounded the kernel coefficient through nine tangential derivatives and \(H\) through seven. The full orthogonal equation has also supplied the gain in (84) without losing derivative orders. We use that gain in a second even projection; no eighth derivative of \(H\) is required, because \(P_\perp\) annihilates \(Hg'\).

The second parity projection. Return to (79). Positive derivatives through order 6 of the even part of \(E\) now obey \[ \|\partial_y^kE^{\rm even}\|_{L_t^2} \le C h^{-3-1/4+C\epsilon},\qquad1\le k\le6. \tag{85}\] Indeed the odd term \(-t|A|^2g'\) still contributes nothing; a positive derivative of \(\mathop{\mathrm{tr}}A^3\) contains a factor bounded by \(h^{-5/4}\); every horizontal metric or drift error contains at least one positive \(y\)-derivative of \(\varphi\), so (84) applies up to order 8. If derivatives instead fall on its coefficient, that coefficient remains \(O(h^{-1+C\epsilon})\) and the same estimate applies to the positive derivative already present on \(\varphi\). The normal drift and nonlinear terms remain \(O(h^{-4+C\epsilon})\) in these norms. This accounts for all terms, including products of derivatives and the superpolynomially small cutoff errors.

Apply the even orthogonal inverse once more. Its value at zero and the zero condition give \(|\partial_y^ku_0|\le C h^{-3-1/4+C\epsilon}\) for \(1\le k\le4\). Equation (83) gives the same estimate for \(\partial_yH\) and \(\partial_y^2H\). Choose \(\epsilon\) so small that every fixed exponent loss \(C\epsilon\) above is less than \(1/8\); decreasing the resulting gain if necessary gives \(\gamma>0\). Covariant derivatives have the asserted bounds as well: the coordinate connection is bounded and its extra terms multiply already controlled first derivatives of \(H\).

Recovering the large-scale curvature derivative. The improved mean-curvature derivatives are now in hand. Rescale the original graph disk by \(h^{-1}\). Its existing \(C^{2,1/2}\) bound is uniform, and the rescaled mean curvature \(hH\) has uniformly bounded \(C^{1,1}\) norm, since its first and second derivatives scale by \(h^2\) and \(h^3\), respectively. Differentiate the prescribed mean-curvature graph equation once and apply interior Schauder estimates. The rescaled graph has a uniform third derivative bound, giving \(|\nabla_\Sigma A|\le Ch^{-2}\) in original variables. All boxes used above have a fixed total fractional shrinkage and all derivative orders were fixed in advance. Repeating at each inner center proves the uniform inner-window assertion. ◻

Proof. Comparison along the inward radius. Let \(s\) denote inward depth in the annulus \(0<s<Q=\sqrt r\). The nonnegative function \[w(s)=g'(s)\int_0^s(g'(t))^{-2} \int_t^\infty(g'(u))^2\,\,\mathrm du\,\,\mathrm dt\] has \(w(0)=0\), \((-\partial_s^2+V_g)w=g'\), and \(|w|+|w'|\le C(1+s)g'\). Thus \(v_*=g-C_0w/r\), with a sufficiently large fixed \(C_0\), is positive on the annulus and satisfies \(\Delta v_*+v_*-v_*^3>0\). Indeed the linear correction contributes \(C_0g'/r\), the inward radial drift is \(-(n-1)v_*'/(r-s)\), and the quadratic correction is bounded by \(C r^{-2}(1+s)^2(g')^2=o(r^{-1}g')\) uniformly on \([0,Q]\). Multiplication by \(1-e^{-cQ}\), with a sufficiently small fixed \(c>0\), keeps a strict subsolution and makes its value less than \(v\) on the inner boundary, by phase decay. On the outer boundary it vanishes. Sliding its amplitude from a small positive number to one proves comparison. To justify the start, \(v>0\) in the compact annulus and the Hopf boundary estimate applies at any outer zero; these and compactness bound the quotient from below. At a first contact a strict subsolution cannot touch in the interior, and the Hopf comparison excludes an outer boundary obstruction. Smaller amplitudes remain strict subsolutions: writing \(\mathcal F(w)=\Delta w+w-w^3\), one has \(\mathcal F(av_*)=a\mathcal F(v_*)+(a-a^3)v_*^3>0\) for \(0<a\le1\).

Inverting \(g\) on \(0\le s\le\log r\) proves (87): the correction is bounded by \(C(1+s)g'(s)/r\), and the extra \(e^{-c\sqrt r}\) term is smaller after division by \(g'(s)\) on that range. Differentiation at the contact point gives \(\partial_Nv(x)\ge g'(0)-C/r>0\). Thus \(x\) is regular, \(N=\nu\) after the sign choice, and \(D(x)\le C/r\). The tangent ball gives the principal-curvature lower bound \(\lambda_i(A_N)\ge-1/r\). The zero-level identity \(H_N=\partial_ND/(2|p|^2)\) and (36) give \(|H_N|\le CD\le C/r\). Summing the principal curvatures then bounds each one from above by \(C/r\), proving the full curvature estimate in (86).

Separating the even source from the boundary contribution. The comparison has established regularity and the \(r^{-1}\) bounds. To improve only the upper trace, place \(x=0\), \(N=e_n\), and use the ambient vertical coordinate \(s\). Work in \(B_P\) with \(P=\alpha\log r\), where \(\alpha>0\) is fixed sufficiently small at the end. The differentiated diffuse estimates give, for every needed fixed order \(k\), \[\|D\|_{C^k(B_P)}\le C_k r^{-1+C\alpha},\qquad \|T\|_{C^k(B_P)}+\|U-s\|_{C^k(B_P)} \le C_k r^{-1/2+C\alpha}.\] These follow by applying unit-ball derivative estimates and the exponential comparison for \(D\) along chains from \(0\); integration of \(T\) gives the estimate on \(U-s\). The multipliers of \(\alpha\) are fixed independently of small \(\alpha\). The identity \(\nabla_pT=-T^2-\nabla^2D/2\) further gives \(\partial_sT=O(r^{-1+C\alpha})\): the replacement of \(p\) by \(e_n\) has error \(|p-e_n||\nabla T|=O(r^{-1+C\alpha})\).

Set \(F=G(s)D\). Conjugating the defect equation and replacing \(vp\) by \(g(s)e_n\) and \(q|p|^2\) by \(G(s)\) yields \[ (-\Delta+2-G(s))F =2G(s)|T(x',0)|^2+\mathcal E, \qquad\|\mathcal E\|_\infty\le Cr^{-3/2+C\alpha}. \tag{88}\] For completeness, each replacement in the drift or potential multiplies a coefficient of size \(r^{-1/2+C\alpha}\) by \(D\) or \(\nabla D\) of size \(r^{-1+C\alpha}\). Replacing \(|T(x',s)|^2\) by \(|T(x',0)|^2\) has the same order, since \(|T|=O(r^{-1/2+C\alpha})\), \(\partial_sT=O(r^{-1+C\alpha})\), and \(|s|\le P\). Logarithmic factors are absorbed by increasing the fixed multiplier of \(\alpha\).

Solve the Dirichlet problem on \(B_P\) by linearity. The solution with source \(2G|T(x',0)|^2\) and zero boundary is even in \(s\) and nonnegative. The zero-boundary solution for \(\mathcal E\) has interior \(C^1\) norm at most \(Cr^{-3/2+C\alpha}\), since the potential \(2-G\) is at least one. The remaining homogeneous solution is \(u_++u_-\), both nonnegative, with boundary values \(F\) on the upper and lower hemispheres, respectively, and zero on the other one. Their harmless equatorial boundary discontinuity is interpreted by bounded Dirichlet solutions; it does not affect interior estimates. Reflection and the maximum principle in the upper half-ball give \(\partial_su_-(0)\le0\); local Harnack and derivative estimates give \(|\partial_su_+(0)|\le Cu_+(0)\).

At \(x_*=(0,P/2)\), positivity of the source solution implies \[u_+(x_*)\le F(x_*)+Cr^{-3/2+C\alpha} \le C(1+P)r^{-1}G(P/2)+Cr^{-3/2+C\alpha}.\] For the last inequality integrate \(D\le2(1-\partial_sU)\) along \([0,P/2+1]\), use (87), and apply unit-scale Harnack to \(D\) to pass from the integral to its value at \(P/2\). This use is valid for \(\alpha<1\), so the interval lies in the range of (87).

Suppressing the upper boundary contribution. Only \(u_+\) can contribute a positive leading normal derivative at the origin. Its value at \(x_*\) is small by the inward comparison; we now transfer that gain to the origin with explicit Poisson bounds. Write \(A_\partial=\int_{\partial B_P\cap\{s>0\}}F\,\,\mathrm dS\). Comparison with potentials one and two gives \[ u_+(0)\le CP^C e^{-P}A_\partial, \qquad u_+(x_*)\ge c_\zeta e^{-(\sqrt2+\zeta)\sqrt5 P/2}A_\partial \quad(\zeta>0). \tag{89}\] For the first inequality, the Poisson density at the center for \(-\Delta+1\) is constant on the sphere. The spherical average of \(e^{x\cdot\theta}\) shows that its size is at most \(CP^Ce^{-P}\). The solution for potential \(2-G\ge1\) is smaller. For the second, the solution for potential \(2-G\le2\) is larger than that for \(-\Delta+2\). Its Dirichlet Poisson density at any upper-hemisphere point \(\xi\) is at least \(c_\zeta e^{-(\sqrt2+\zeta)|x_*-\xi|}\). Here are quantitative details of this last bound. Start the Dirichlet Green function a fixed distance from \(x_*\), with a fixed positive lower bound obtained from a fixed-radius ball. Along the segment towards \(\xi\) insert a cylinder of fixed large cross-sectional radius, chosen so that \(\sqrt{2+\lambda_1}<\sqrt2+\zeta\). Its first transverse Dirichlet mode times the longitudinal hyperbolic-sine solution propagates the lower bound with that exponential rate. Stop a fixed distance before \(\xi\). Such a cylinder fits in \(B_P\) except for fixed initial and final portions: \(x_*\) is at distance \(P/2\) from the boundary and the incidence angle at \(\xi\) has a cosine bounded below uniformly. Harnack on those fixed portions and a tangent-ball barrier at \(\xi\) transfer the bound to the inward normal derivative. All constants depend on \(\zeta\) but not \(P\). Finally \(|x_*-\xi|\le\sqrt5 P/2\), proving (89).

Combining the two bounds with \(G(P/2)\le Ce^{-\sqrt2 P/2}\) gives \[u_+(0)\le CP^C r^{-1} e^{[(\sqrt2+\zeta)\sqrt5/2-1-\sqrt2/2]P} +CP^C r^{-3/2+C\alpha} e^{[(\sqrt2+\zeta)\sqrt5/2-1]P}.\] Since \(\sqrt{10}/2-1-\sqrt2/2<0\), choose \(\zeta\) so the first exponent is negative, then choose \(\alpha>0\) so the second term still has a power better than \(r^{-1}\). Polynomial factors in \(P\) can be absorbed by slightly decreasing a resulting gain \(\gamma>0\). The source solution has zero derivative at \(0\), the lower-hemisphere solution has nonpositive derivative there, and the error solution is smaller still. Thus \(F_s(0)\le Cr^{-1-\gamma}\). Since \(G(0)=1\), \(G'(0)=0\), and \(|p(0)|^2=1-O(r^{-1})\), the zero-level identity for \(H_N\) proves (86). ◻

Proof. Semiconcavity, including diagonal competitors. We first justify the weak interpretation, independently of the differentiations to follow. On a compact regular coordinate patch there are uniform bounds \(0<c\le d\le C\): local graph curvature gives the lower bound, and one fixed off-diagonal positive-angle competitor gives the upper bound after shrinking the patch. Relevant competitors of value at most \(2C\) have bounded length. Those of length at least \(\varepsilon>0\) have angle \(s\ge\varepsilon/(2C)\), by \(\Gamma\ge1/s\). With the second foot fixed, these are smooth branches in the base variable, uniformly bounded in \(C^2\); no regularity of the second foot is needed here.

For shorter chords the second point is on the same regular local graph. In coordinates \(X(u)\) write it as \(X(u+t\omega)\), with \(|\omega|=1\) and \(t\) small. Such chords use \(\Gamma=1/s\). The reciprocal branch is \[\frac{|\nu(u)\cdot[X(u+t\omega)-X(u)]|} {|X(u+t\omega)-X(u)|^2}.\] Both numerator before absolute value and denominator vanish to second order in \(t\), since \(DX\) is tangent. Cancelling \(t^2\) by the integral Taylor formula gives a \(C^2\) function through \(t=0\), uniformly in the compact direction parameter. On relevant branches its absolute value is bounded below by \(1/(2C)\), so a smooth sign can be chosen and taking its reciprocal preserves the uniform \(C^2\) bound. The diagonal limits are legitimate upper branches for \(d\), since they are limits of competitors. Compactness of the remaining parameter sets shows that every minimizing sequence has either an off-diagonal or a diagonal realizing parameter and that \(d\) is the infimum of a family with locally uniform upper Hessian bound. Subtracting the corresponding fixed quadratic function makes each branch concave and hence also their infimum concave. This proves local semiconcavity.

At almost every point there is an Alexandrov second-order expansion. At such a point any smooth active upper branch has the same first derivative and an upper bound for the Hessian of \(d\). Its trace therefore bounds the absolutely continuous Laplacian. The singular part is a nonpositive measure by semiconcavity, in coordinates and therefore for the surface Laplacian as well. It suffices to compute the following upper touching branches.

The weak interpretation is now established. We next compute smooth upper touching branches, treating the unmodified penalty, diagonal limits, and the transition to the plateau in that order. The final step will transfer the positive plateau contribution to its receiving foot.

A tangent phase ball at every second foot. For an off-diagonal contact choose \(N=\pm\nu(z)\) so \(N\cdot e=s>0\). The open region \[\{z+\ell'e':N\cdot e'>0,\ \ell'\Gamma(N\cdot e')<d\}\] contains no zeros. After translation, rotation, and division by \(d\), its boundary is a fixed smooth embedded compact radial surface. Near the origin it agrees with the sphere \(|x|^2=x_n\), because there \(\Gamma(s)=1/s\). On the remainder it is the smooth positive radial graph \(1/\Gamma\). It has an inward rolling ball of a fixed positive radius \(c_\Gamma\): local \(C^2\) graph bounds supply the balls near each point, and compactness and embeddedness allow a uniform decrease so those balls avoid the rest of the boundary. Hence at the second foot \(y\) there is a zero-free tangent ball of radius \(c_\Gamma d\). Lemma 25 shows that the foot is regular, \(a(y)+D(y)\le C_\Gamma/d\), and its inward trace satisfies \(H_{N_y}(y)\le C_\Gamma d^{-1-\gamma}\).

Off-diagonal contacts with the unmodified penalty. First suppose \(s\le1-\delta_1\), so \(\Gamma=1/s\). Write \(r=d/2\), \(b=e-sN\), and \(M=\mathrm{Id}+rA_N\). Then \(\ell=2rs\) and the inward normal at the second foot is \(N_y=N-2se\). Reflect an orthonormal frame \(v_i\) of \(T_z\Sigma\) across \(e^\perp\) to obtain \(v_i'=v_i-2(e\cdot v_i)e\) in \(T_y\Sigma\). Move \(z\) and \(y\) on the respective geodesics with these initial velocities. The resulting chord branches touch \(d\) from above. First and second differentiation gives \[ \begin{split} \nabla_\Sigma r={}&-Mb/s,\\ \Delta_\Sigma d\le{}&a^2d+ \left(2-\frac2{1+r\max_i|\lambda_i|}\right) \frac{|\nabla_\Sigma d|^2}{d}\\ &+\frac{H_{N_y}(y)-(1-2s^2)H_N(z) -\ell b\cdot\nabla_\Sigma H_N(z)}{s^2}. \end{split} \tag{94}\] Here is the algebra, including the factor of the gradient term. Put \(w=y-z\) and \(w_i=v_i'-v_i=-2(e\cdot v_i)e\) and differentiate \(|w|^2/2-rN\cdot w\), which vanishes at contact and is nonnegative for a competitor. The summed second derivative bounds \((\Delta r)s\ell\) by \[\begin{split} &\sum_i|w_i|^2+(w-rN)\cdot(-H_{N_y}N_y+H_NN) -2r\sum_iw_i\cdot A_Nv_i\\ &\hspace{8mm}-rw\cdot(\nabla H_N-a^2N) -2\sum_i r_i\bigl(w_i\cdot N+w\cdot A_Nv_i\bigr). \end{split}\] Use \(w-rN=-rN_y\) and the first derivative identity. The algebraic terms besides \(a^2\) and the traces reduce to \(4s^2|\nabla r|^2+4s b\cdot\nabla r\). By (72), \(0\le M\le (1+r\max_i|\lambda_i|)\mathrm{Id}\), so \[b\cdot\nabla r=-b\cdot Mb/s \le-\frac{s|\nabla r|^2}{1+r\max_i|\lambda_i|}.\] Dividing the preceding second derivative expression by \(rs^2\) therefore gives exactly (94). The identity \(\Delta_\Sigma N=\nabla_\Sigma H_N-a^2N\) fixes all trace signs.

We bound its last quotient uniformly, including almost tangential contacts. Choose a small fixed \(\gamma'>0\) compared with the gains in Lemmas 24 and 25. If \(s\ge d^{-\gamma'}\), the second-foot upper trace, the base bounds \(|H|\le Ch^{-2}\) and \(|\nabla H|\le Ch^{-3-\gamma}\), and \(h=d^{1-o(1)}\) bound that quotient above by \(C_\Gamma d^{-1-\gamma''}\) for some fixed \(\gamma''>0\). If \(s<d^{-\gamma'}\), then \(\ell=ds\) lies inside the isolated graph. The two normal orientations agree by continuation there. In its tangent-plane coordinates the displacement is exactly \(\ell b\); Taylor’s formula gives \[H_{N_y}(y)-H_N(z)-\ell b\cdot\nabla H_N(z) =O(\ell^2h^{-3-\gamma}).\] The coordinate Hessian obeys this bound as well, since its connection correction multiplies the already controlled gradient. Dividing by \(s^2\) and adding \(2H_N(z)\) proves the same upper error after possibly decreasing \(\gamma''\).

The coefficient in parentheses in (94) is at most one, because \(r\max|\lambda_i|\le1\). It is \(O(ad)\) if \(ad\) is small. Also \(|b|\ge\sqrt{1-(1-\delta_1)^2}>0\). If \(ad\) is sufficiently small then the first derivative identity gives \(|\nabla d|\ge c_\Gamma\). Otherwise \(a^2d\ge c_\Gamma/d\). In either case the error \(C d^{-1-\gamma''}\) can be absorbed into \(C d^{-\gamma''}(a^2d+|\nabla d|^2/d)\). This proves (91) in the present range.

For the gradient bound, if \(\ell\ge ch\), then \(1/s=d/\ell\le Cd/h\) and \(\|M\|\le2\), giving \(|\nabla d|\le Cd/h\). For \(\ell<ch\) both points lie on the isolated graph. Expanding its normal and projecting tangentially at the base yields \[-2s b=\ell A_Nb+O(\ell^2/h^2),\] using \(|\nabla A|\le Ch^{-2}\) and \(|A|\le Ch^{-1}\). Thus \(|Mb|\le Cs d^2/h^2\), and \(|\nabla d|\le C d^2/h^2\). Both estimates imply (90).

Diagonal contacts. At a diagonal contact the curvature itself is the active constraint. Choose \(N\) so \(S=-A_N\) has top positive eigenvalue \(\lambda=2/d\), and let \(v_1\) be an active top eigenvector. Every eigenvalue \(\lambda_j\) of \(S\) is between \(-\lambda\) and \(\lambda\). For a unit vector \(v\) in the top eigenspace \(V\), expansion along the geodesic with signed length \(L\) gives \[\frac{2N\cdot(y-z)}{|y-z|^2} =\lambda+\frac13\nabla_vS(v,v)L+O(L^2).\] This reciprocal cannot exceed \(2/d=\lambda\) for either sign of \(L\), so \(\nabla_vS(v,v)=0\). Codazzi makes \(\nabla S\) totally symmetric, and polarization implies that it vanishes on triples in \(V\).

Touch \(d\) from above by \(2/S(v_1,v_1)\) using a unit field extension. At the point choose its derivatives, in a principal frame, to be \[\langle\nabla_i v_1,v_j\rangle =\frac{\nabla_iS_{1j}}{\lambda-\lambda_j} \quad(\lambda_j<\lambda), \qquad \langle\nabla_i v_1,v_j\rangle=0\quad(v_j\in V).\] A smooth local unit extension with these specified first derivatives exists; its second derivatives are unconstrained apart from the unit length condition, and their tangential components drop out of the Laplacian of \(S(v_1,v_1)\) at a top eigenvector. Thus that Laplacian is \[(\Delta S)_{11} +2\sum_{\lambda_j<\lambda} \frac{|\nabla S_{1j}|^2}{\lambda-\lambda_j}.\] The Euclidean hypersurface identity \[ \Delta S=\nabla^2\mathop{\mathrm{tr}}S+(\mathop{\mathrm{tr}}S)S^2-a^2S \tag{95}\] follows by commuting the two derivatives in Codazzi and applying the Gauss equation. This is the Simons curvature calculation (Simons 1968; Schoen et al. 1975), with the mean-curvature terms retained because the sheet need not be minimal. More explicitly the Ricci contraction is \(((\mathop{\mathrm{tr}}S)S-S^2)_{ip}S_{pj}\) and the remaining curvature contraction is \(S^3_{ij}-a^2S_{ij}\), giving the stated identity. Also \(\lambda-\lambda_j\le2\lambda\), and Codazzi gives \(\nabla_jS_{11}=\nabla_1S_{j1}\), which is zero for \(v_j\in V\). Consequently the denominator \(f=S(v_1,v_1)\) satisfies \[\Delta f\ge-a^2\lambda -C(|\nabla^2H|+a^2|H|)+|\nabla f|^2/\lambda.\] Differentiating \(2/f\) now gives \[\Delta d\le a^2d+\frac{|\nabla d|^2}{d} +C d^2(|\nabla^2H|+a^2|H|).\] The last term is \(C d^{-\gamma}(a^2d)\) by Lemma 24, \(h=d^{1-o(1)}\), and \(a^2d\ge4/d\). This proves (91). A diagonal contact always has \(ad\ge2\), so the small-\(ad\) assertion makes no claim here. Finally \(|\nabla d|\le Cd^2|\nabla A|\le Cd^2/h^2\) proves (90) for these contacts too.

The angular transition and the plateau. The unmodified and diagonal cases satisfy the claimed inequality. It remains to handle \(s\ge1-\delta_1\), where the penalty changes the variation at the second foot. Set \[b=e-sN,\qquad c_1=\Gamma-s\Gamma',\qquad Z=c_1e+\Gamma'N.\] First variation in the second foot shows that its inward normal is \(N_y=-Z/|Z|\). Rotate the tangent frame at \(z\) by the small plane rotation taking \(N\) to \(Z/|Z|\), to obtain \(v_i'\) at \(y\). With \(w_i=v_i'-v_i\), elementary plane-rotation formulas give \[w_i=O(|b|)N+O(|b|^2),\qquad P_{e^\perp}w_i=O(|b|^2),\qquad (N-se)\cdot w_i=O(|b|^3).\] The constants are uniform for the fixed penalty; in fact \(|Z|=1+O(\delta_1)\) and the rotation is uniformly small. At \(b=0\) use the identity rotation. Differentiation of \(\ell\Gamma(s)\) gives the first derivative and the following upper trace: \[ \begin{split} \nabla d={}&(-c_1\mathrm{Id}+\ell\Gamma'A_N)b,\\ \Delta d\le{}&|Z|H_{N_y}+H_N Z\cdot N +\ell\Gamma' b\cdot\nabla H_N-\ell\Gamma's a^2\\ &+\sum_i\left\{ \frac{c_1}{\ell}|P_{e^\perp}w_i|^2 +2\Gamma' w_i\cdot A_Nv_i +\ell\Gamma''\left[b\cdot A_Nv_i +\frac{(N-se)\cdot w_i}{\ell}\right]^2\right\}. \end{split} \tag{96}\] This can also be derived directly from \(\,\mathrm d\ell=e\cdot\,\mathrm dw\) and \(\,\mathrm ds=\,\mathrm dN\cdot e+(N-se)\cdot\,\mathrm dw/\ell\): the Hessian in \(w\) contributes the first term in braces, the two mixed normal derivatives the second, the second derivative of \(\Gamma\) the square, and \(\Delta N=\nabla H_N-a^2N\) gives the two normal-variation terms on the first line of the trace. The accelerations of the two geodesic frames give the two trace terms.

Since \(\Gamma''|b|^2\le C\varepsilon_1\), the algebraic terms, including \(-\ell\Gamma'sa^2\), are at most \[(1+C(\varepsilon_1+\delta_1))a^2d +Ca|b|^2+C|b|^4/d.\] For clarity the leading coefficient is bounded by one up to \(O(\delta_1)\) because \(-s\Gamma'/\Gamma\le1+C\delta_1\); the \(\Gamma''\) square contributes \(C\varepsilon_1a^2d\) and \(C\varepsilon_1|b|^4/d\), and the rotation gives the remaining two terms. Young’s inequality bounds \(Ca|b|^2\) by any fixed small multiple of \(a^2d\) plus a fixed multiple of \(|b|^4/d\). If \(ad\) is small, the first derivative formula bounds \(|\nabla d|\) below by a fixed multiple of \(|b|\). If it is not small, \(|b|^4/d\) is at most \(C\delta_1^2a^2d\). Choosing \(\varepsilon_1\) and then \(\delta_1\) small enough for the prescribed tolerance therefore bounds all these terms by \((1+\eta_0)a^2d+\eta_0|\nabla d|^2/d\), with room to decrease the initial tolerance for the subsequent errors. In the transition interval before the plateau, \(|b|\) has a fixed positive lower bound depending on \(\delta_2\). The trace terms in (96) are \(O(d^{-1-\gamma})\) from above by the base and second-foot estimates. They are absorbed exactly as in (94), proving (91), including its small-\(ad\) statement. The gradient formula, \(\ell\le d\), and \(da\le2\sqrt m\) give a fixed bound in this range, hence (90).

On the plateau \(\Gamma'=\Gamma''=0\), so \(\nabla d=-G_1b\), \(Z=G_1e\), \(N_y=-e\). The only algebraic remainder is \(C|b|^4/d\), bounded by \(C\delta_2|\nabla d|^2/d\). The traces are \(G_1H_{N_y}+G_1sH_N\le G_1(H_{N_y,+}+|H_N|)\). Decreasing \(\delta_2\) if necessary proves (92). At the endpoints of the ranges the jets match, since the constructed penalty is smooth; either computation thus has the same limiting validity.

The differential inequalities have now been proved for every active contact type. It remains to justify their integrated use: a positive plateau contribution at a receiving foot must be counted with its inverse area factor, and without an additional multiplicity.

The inverse Jacobian and positive receiving multiplicity. We prove the transfer statement in the direction from receiving feet to bases; this direction determines the Jacobian below. For a plateau pair, \(z=y+\ell N_y\) and \(|N_y\cdot\nu(z)|=s\ge1-\delta_2\). Fix a local regular sheet through \(z\). The equation that \(y+tN_y\) lies on this sheet has nonzero \(t\) derivative, so the implicit function theorem supplies a smooth inverse parametrization \(z(y)=y+\ell(y)N_y\) near this pair. Projecting its derivative onto \(T_y\Sigma\) removes \((\,\mathrm d\ell)N_y\) and leaves \(\mathrm{Id}+\ell A_{N_y}\). Orthogonal projection from \(T_z\Sigma\) to \(T_y\Sigma\) has area factor \(s\). It follows that the area Jacobian of the inverse map is exactly the quotient in (93). The second-foot bound \(a(y)\le C_\Gamma/d(z)\) therefore bounds this quotient uniformly. Expanding the determinant when \(a(y)\ell\) is small gives the more precise estimate stated. The reverse chord is normal at \(y\), so it is a competitor of value \(G_1\ell\) there and proves \(d(y)\le d(z)\).

At a foot with \(H_{N_y}>0\) the orientation \(N_y\) is unique. All bases sending positive charge to this foot must lie on its same normal ray. Suppose two existed, at lengths \(0<\ell_1<\ell_2\). For the farther base \(z_2\), the nearer base \(z_1\) is a competing zero with chord direction exactly equal to the direction towards \(y\) and length \(\ell_2-\ell_1\). It has the same angle penalty and strictly smaller objective than the alleged active chord to \(y\), a contradiction. Thus every positively charged foot has at most one inverse base.

There is no hidden diffeomorphism assumption here. Cover the regular pair relation by countably many of the preceding inverse charts. Active parameters on compact coordinate pieces form a closed compact-valued relation; a measurable selection can be obtained there and pieced together over a countable exhaustion. Restrict the selected pairs to disjoint measurable pieces in those charts. The area formula on each inverse chart gives exactly its Jacobian; its critical image has zero surface measure. Summing the formula over the selected pieces adds no multiplicity for positive charges by the ray argument. In explicit terms, for a nonnegative measurable integrand \(F(z,y)\) supported on positively charged selected pairs, \[\int F(z,y(z))\,\,\mathrm dS_z =\int F(z(y),y)J_{y\to z}\,\,\mathrm dS_y,\] where the right-hand integral is over the selected positive feet, with null critical images omitted on the left. This proves the claimed transfer for arbitrary nonnegative weights, not only constant ones, and completes the proposition. ◻

Proof. Use the tangent balls of Lemma 20 and Lemma 25. The proof of Proposition 26 supplies the corresponding tangent balls at second feet, with radius measured by the source reach \(d(z)\). Exponential comparison in (36) bounds \(D\) below by a positive constant depending only on \(\delta,L_0\) within distance \(10L_0\) of \(X\). Combining it with the appropriate upper bound \(C_\Gamma/d(z)\) gives the asserted reach bound. If \(X\) is empty the collar assertion has no points to consider. ◻

Proof. It suffices to treat \(0<\varepsilon<1\), since decreasing \(\varepsilon\) strengthens the conclusions. The two-sided comparison with \(D\) follows from the local Harnack bound in (36), the positive lower bound for \(q\) in a fixed small neighborhood of a zero, and its upper bound on a fixed segment. The mean-curvature estimate is the finite-window conclusion of Lemma 13. To prove the additional curvature estimate, and to identify the common limiting quantities, consider the normalized limits of Lemma 10, with \(z=0\), \(\nu(z)=e_n\), and normalization by \(D(0)\) for \(D\) and by \(\sqrt{D(0)}\) for \(T\). Write these limits as \(\mathcal D,\mathcal T\). Then \(\mathcal D(0)=1\), \(\mathcal T\) is a horizontal harmonic tensor independent of \(s=x_n\), and \[F(x',s)=G(s)\mathcal D(x',s),\qquad (-\Delta+2-G)F=2G(s)|\mathcal T(x')|^2.\] We use the nonnegative source part of Lemma 11 to compare this mass with curvature. The function \(|\mathcal T|^2\) is horizontally subharmonic. In a product cylinder \(B_R^m\times(-R,R)\) the positive Dirichlet Green kernel, evaluated at \((0,s)\), is radial in the horizontal source variable. Horizontal spherical averaging therefore shows that its integral against \(2G|\mathcal T|^2\) is at least its integral against \(2G|\mathcal T(0)|^2\). The zero-boundary source solution is bounded above by \(F\). As \(R\to\infty\), the latter constant-horizontal source solutions increase to \(|\mathcal T(0)|^2 f(s)\), where \[(-\partial_s^2+2-G)f=2G,\qquad f\ge0,\] with \(f\) bounded. To justify this identification, comparison with the constant \(2\) bounds the exhaustions; every bounded entire solution with this source is unique, since the potential is at least \(1\), and horizontal translations then force independence of \(x'\). The uniqueness follows by comparison on balls with the growing exponential barriers for \(-\Delta+1\). Since \((-\partial_s^2+2-G)G=2G^2\), integration against \(G\) gives \[\frac1\sigma\int_\mathbb RG(s)^2F(0,s)\,\,\mathrm ds \ge |\mathcal T(0)|^2.\] All integrations are legitimate by boundedness of \(f\) and exponential decay of \(G\); one may first insert compact cutoffs.

These limiting inequalities imply their stated finite-window versions. Here is the uniformity argument for the curvature assertion, where an additive limiting error would be insufficient. If no fixed interval and threshold worked, choose \(P_j\to\infty\), \(D(z_j)\to0\), and \(I_{P_j}<(1-\varepsilon)a(z_j)^2\). The lower local mass bound gives \(a(z_j)^2/D(z_j)\ge c/(1-\varepsilon)>0\). Subsequence convergence on every fixed interval would contradict the preceding limiting inequality. Here \(a^2/D\) converges to \(|\mathcal T(0)|^2\). Choose the interval large enough also for Lemma 13 with coefficient \(1/(2\sqrt2)+\varepsilon\), and then take the smaller of the two defect thresholds. Its nonnegative integrand preserves that mean-curvature bound on every larger fixed interval. This proves all assertions with the stated order of choices. ◻

Proof. Smooth single-profile convergence on every fixed ball follows from (36). In the normalized limit used above, \(\nabla_\Sigma H/D(z)\) converges to \(\nabla_{x'}F_s(0)/2\). The even source part of \(F\) contributes zero to this derivative. For a positive homogeneous extremal \(F=e^{b\cdot x'}f(s/\sqrt2)\), the explicit profiles in Lemma 11 give \[|\nabla_{x'}F_s(0)|\le C|b|F(0),\qquad |b|\le\sqrt{3/2}.\] The average of \(e^{b\cdot x'}\) on \(B_{S/2}^m\) is at least \(1\), and tends to infinity uniformly when \(|b|\) is bounded below by any positive number. First restricting to \(|b|<\eta\), then taking \(S\) large on its complement, proves, with any prescribed \(\zeta>0\), \[\frac12|\nabla_{x'}F_s(0)| \le \zeta\fint_{B_{S/2}^m}F(x',0)\,\,\mathrm dx'.\] The same inequality holds for positive mixtures by the triangle inequality; the nonnegative even source can only enlarge its right side. Choose \(\zeta\) in terms of \(\varepsilon\) and apply Cauchy–Schwarz. Since \(F(x',0)=\mathcal D(x',0)\), this proves the limiting version of (98). Smooth normalized convergence and the uniformly positive local mass of \(\mathcal D^2\) allow a fixed strict relaxation. The graph area elements converge to \(\,\mathrm dx'\), and the disk \(B_{S/2}^m\) lies in the surface ball used on the right. A contradiction sequence now gives a uniform positive threshold \(\delta\). ◻

Proof. On \(|s|a<\alpha\), \(\Phi\) is a local diffeomorphism and \(J>0\). A fiber restricted to the compact lifted support is discrete and compact, hence finite. A sufficiently small physical neighborhood has only these nonzero inverse branches: otherwise a sequence of additional branches has a limit in the same compact support and contradicts local invertibility. Consequently the squared sum is locally a finite sum. Its square root is Lipschitz, including at its zero set, and almost everywhere \[\sum_j|\nabla\xi_j|^2-|\nabla\xi|^2 =\sum_j|\nabla\xi_j|^2- \frac{|\sum_j\xi_j\nabla\xi_j|^2}{\sum_j\xi_j^2} =\mathcal V\ge0.\] At points where all terms vanish, their gradients vanish almost everywhere, so the same equality holds with value zero. The potential terms add exactly, irrespective of their sign. The area formula for \(\Phi\) thus proves the assertion without an injective global tube.

For any lifted amplitude \(G f\), expand its normal derivative and integrate the cross term \(2GG'ff_sJ\) by parts. Using \(G''=(3g^2-1)G\) gives precisely (99), with \(f=\beta\psi\). There are no artificial chart boundaries in this computation: the amplitude is globally defined on the lifted surface, while local inverse charts are used only to justify its pushdown. Finite even normal caps may be removed whenever the displayed terms are dominated by a polynomial in \(|s|\) times \(G^2\); this is the case in the applications below. At \(a=0\) take \(\beta=1\) and use the same capped limit. ◻

Proof. Use Lemma 30. On the support of \(\beta\) the pullback tangential metric differs from the surface metric by \(O(\alpha)\). Uniform derivative estimates give \(|\nabla\beta|\le C_\alpha(1+|s|)\); derivatives of \(a=|A|\) are understood weakly at its zero set. All terms containing a derivative of \(\beta\) are bounded, after a Young inequality with any fixed small parameter, by \(C_\alpha e^{-c_\alpha/a}\psi^2\). Thus the gradient part is at most \[(1+C\alpha+\eta)\sigma|\nabla_\Sigma\psi|^2 +C_{\alpha,\eta}e^{-c_\alpha/a}\psi^2.\] The exponential remainder is \(o(D)\) uniformly as \(D\to0\), since \(a^2\le CD\).

The elementary-symmetric expansion of \(J\) and evenness give \[-\int\beta^2J_sGG'\,\,\mathrm ds =\frac\sigma2(H^2-a^2)+o(D).\] Indeed the constant term \(H\) in \(J_s\) integrates to zero, the linear term is \((H^2-a^2)s\), and \(-\int sGG'=\sigma/2\). Terms of order at least two in \(sA\) are either odd under the integral or bounded by \(Ca^3=o(D)\). The cap errors have the preceding exponential bound, and \(H^2=o(D)\).

The potential contribution is nonpositive everywhere: \(U(z)=0\) and \(|\nabla U|\le1\) imply \(|U(z+s\nu)|\le|s|\) and hence \(v(z+s\nu)^2\le g(s)^2\). We quantify its gain. On the two normal rays define \(U_\pm(r)=\pm U(z\pm r\nu)\) for \(r\ge0\). Then \[r-U_\pm(r)\ge\frac12\int_0^rD(z\pm t\nu)\,\,\mathrm dt.\] Indeed \(U_\pm'(r)=p(z\pm r\nu)\cdot\nu \le\sqrt{1-D(z\pm r\nu)}\le1-D(z\pm r\nu)/2\). This writes the same deficit inequality on both sides without changing its sign convention. Moreover, on every fixed segment, \[ |p(z+s\nu)-\nu|+|U(z+s\nu)-s|\le C_P D(z),\qquad |s|\le P. \tag{101}\] To prove it, initially \(|p(z)-\nu|\le CD(z)\), and along the segment \[|T\nu|\le|Tp|+|T||p-\nu| \le C_P D(z)+C_P\sqrt{D(z)}|p-\nu|.\] A Gronwall estimate gives (101). Here \(Tp=-\nabla D/2\) and (36) were used.

For a fixed segment \([-Q,Q]\), (101) gives \(p\cdot\nu>0\) once the threshold is small enough. Thus \(U(s)\) has the sign of \(s\) on both half-segments. After division by \(D(z)\) and passage to a normalized limit, the potential gain \(3(g^2-v^2)G^2\) is at least \[-\frac12(G^3)'(s)\int_0^s\mathcal D(0,t)\,\,\mathrm dt.\] This expression is nonnegative on both half-lines. Integrating and using Tonelli, its total mass is \(\tfrac12\int_\mathbb RG^3\mathcal D\,\,\mathrm ds\). To make the finite-window order explicit, first fix \(P\) and put \(h_P(s)=\int_0^s\mathbf1_{\{|t|\le P\}}\mathcal D(0,t)\,\,\mathrm dt\). For \(Q\ge P\), integration by parts gives exactly \[-\frac12\int_{-Q}^{Q}(G^3)'h_P\,\,\mathrm ds =\frac12\int_{-P}^{P}G^3\mathcal D\,\,\mathrm ds -\frac{G(Q)^3}{2}\int_{-P}^{P}\mathcal D\,\,\mathrm ds.\] Local Harnack bounds the last integral by \(C_P\), uniformly for normalized limits. Choose \(Q\) large after \(P\) so the loss is as small as prescribed, then decrease the defect threshold to use compactness on \([-Q,Q]\). This threshold also ensures \(\beta=1\) on that segment. Since \(I_P/D\ge c\), for any \(\eta>0\) this gives \[\int 3(g^2-v(\Phi)^2)G^2\beta^2J\,\,\mathrm ds \ge(1-\eta)\frac\sigma2 I_P\] when the defect threshold is sufficiently small. Fix \(\eta\), then \(\alpha\) and the threshold, in that order, to absorb the \(o(D)\) terms using \(I_P\ge cD\). Physical stability and the upper bound by the lifted form now prove (100). All normal intervals used here are fixed before the final defect threshold: choose \(P\), then the larger extraction interval \(Q\), and then decrease the threshold. No later surface collar or surface-test amplitude enters those choices. ◻

Proof. Let \(v_0\) be the bounded positive half-line function supplied by Lemma 23 with initial endpoint \(0\). It satisfies \((-\partial_t^2+V_g)v_0=1\), \(0<c\le v_0\le C\), and \(|v_0'|+|v_0''|\le C\). Its trace at \(0\) is positive; the functions to which we apply the inverse, rather than this comparison function, have zero trace. Ground-state substitution, first for compact zero-trace functions, gives \[\int\bigl(|\nabla f|^2+V_g f^2\bigr) =\int v_0^2|\nabla(f/v_0)|^2+\int v_0^{-1}f^2 \ge C^{-1}\|f\|_2^2.\] The energy inverse therefore exists for compactly supported sources. For a source in a unit block, energy estimates followed by interior and flat-boundary elliptic estimates and Sobolev bootstrapping bound its \(W^{2,p_0}\) norm locally by the source’s \(L^{p_0}\) norm. These bounds are uniform in the block location; only a fixed number of bootstraps is required because \(p_0>n\) is fixed.

Off a fixed enlargement of that block, compare its absolute value with \(Cv_0(t)e^{-c_1 r}\), where \(r\) is distance to the block center. For sufficiently small fixed \(c_1>0\), this is a supersolution: \(L_gv_0=1\) dominates the terms bounded by \(C(c_1+c_1^2+c_1/r)v_0\). Comparison is valid after division by \(v_0\), since the divided operator has positive zeroth coefficient \(v_0^{-1}\). On unbounded domains it follows by exhaustion using outer growing exponential barriers; alternatively, the energy solution’s vanishing exterior \(L^2\) norm and local estimates give the boundary condition at infinity. Local elliptic estimates then give exponential decay of the \(W^{2,p_0}\) norms as well. Partition an arbitrary source into disjoint unit blocks and sum these estimates. The sum converges locally absolutely because \(e^{b|y'|}\le e^{b|y|}e^{b|y-y'|}\) and \(b<c_1/2\). It provides the stated inverse and norm bound. Here is an explicit uniqueness argument for the full weighted class, including directions with \(t\to\infty\). Let \(R_*(x)=\sqrt{1+|x|^2}\), and write \(C_0=\sup v_0\), \(C_1=\sup|v_0'|\). Since \(|\nabla R_*|\le1\) and \(\Delta R_*\le n\), \[L_g(v_0e^{cR_*}) \ge e^{cR_*}\bigl[1-2cC_1-ncC_0-c^2C_0\bigr].\] Choose \(c_*>0\) so the bracket is at least \(1/2\), and decrease \(b_0\) to be at most \(\min(c_1/4,c_*/2)\). A homogeneous solution in the weighted class satisfies \(|u(y,t)|\le C_u e^{b|y|}\) by local Sobolev embedding. On the hemisphere \(|x|=R\), the barrier \(\eta v_0e^{c_*R_*}\) dominates \(|u|\) for every sufficiently large \(R\), since \(c_*>b\). On its flat boundary \(u=0\). The divided maximum principle on half-balls, for both signs of \(u\), therefore gives \(|u|\le\eta v_0e^{c_*R_*}\) everywhere. Let \(\eta\downarrow0\). ◻

Proof of Proposition 32. Fix \(z\) and write \(B=J_0(z)+E(z)\). We need only treat \(B\to0\). For the first inequality other values are trivial. For the second, if \(E\le J_0\) there is nothing to prove; if \(B\) is bounded below and \(E>J_0\), then \(d\) is uniformly bounded and \(D\) is uniformly bounded below. The latter assertion follows from single-profile compactness: \(D(z_j)\to0\) would make \(U\) converge to a linear coordinate on every fixed ball, forcing \(d(z_j)\to\infty\) by the definition of the penalty. This is also Lemma 20.

Step 1: ordered sheets on the comparison scale. Translate and rotate so \(z=0\), \(\nu(z)=e_n\). Put \(P_0=A_0\log(1/B)\), where \(A_0\) will be large and fixed. After \(A_0\) is chosen, take \(\lambda\) so small that \(10\lambda A_0<1/2\). Every zero in \(B_{10P_0}\) is then good, and \[|A|^2\le J_0(0)e^{10\lambda P_0}\le B^{1/2}.\] Indeed a bad zero there would give \(J_0(0)\ge B^{10\lambda A_0}>B\). At every such good zero \(|\nabla v|\ge\sqrt{3/8}\), so uniform elliptic estimates give uniform graph smoothness on a fixed scale.

The base sheet extends over a disk of radius at least \(2P_0\) in its tangent plane. To see this directly, lift radial segments by tangent projection. Along a lift of length \(O(P_0)\) the normal changes by at most \(O(P_0B^{1/4})=o(1)\), so the projection remains invertible and the lift length remains controlled. All lifts stay inside the regular larger ball, and any terminal limit is regular. Continuation and local uniqueness give a smooth graph; the same argument works for sheets based in a fixed small fraction of \(B_{P_0}\). Their heights differ from their tangent planes by \(O(P_0^2B^{1/4})=o(1)\).

Here is the ordering argument with the length scales recorded. Fix a small angular parameter \(\theta>0\). A sheet based within \(\theta P_0\) has unoriented normal within \(C\theta+o(1)\) of \(e_n\). Otherwise its almost affine disk of radius comparable to \(P_0\) crosses the base disk transversally: along a line in the direction of its tilt, its signed height relative to the base changes sign, with magnitude bounded below by a constant times the tilt times \(P_0\), while both graph errors are \(o(1)\). The intermediate value theorem would give an intersection, impossible for two different regular zero sheets. Thus all pieces in a smaller box are ordered graphs over one horizontal disk. There are finitely many in each closed inner box: infinitely many would accumulate at a regular zero, where the fixed local graph property permits only one piece. Descriptions on overlapping disks agree or are disjoint.

Let the first graph above the base have positive-axis intercept less than \(\theta P_0/10\), if such a graph exists, and let \(\ell\) be its shortest distance to \(0\). Its foot normal \(n'\) satisfies \[ |n'-e_n|\le C\frac{\ell+o(1)}{P_0},\qquad \{n'\cdot x=\ell\}\text{ is its tangent plane at the foot}. \tag{103}\] For this stronger bound, apply the preceding nonintersection argument out to the full horizontal scale comparable to \(P_0\), now using its height \(\ell\) instead of \(\theta P_0\). The minimizing foot is interior because the horizontal chart radius greatly exceeds its intercept. Choosing \(\theta\) small puts the chord in the constant part of \(\Gamma\), and hence \(d(0)\le G_1\ell\).

Choose box fractions \(c_1=c'_0\theta\), \(c_2=c_1/C_0\), with \(c'_0\) sufficiently small and \(C_0\) sufficiently large dimensional constants. These two constants can be fixed before \(\theta\) is decreased further. In \(|y|_\infty\le10c_1P_0\) the adjacent height differs from \(\ell\) by at most a small fixed fraction of \(\ell\), plus \(o(1)\), by (103). This box contains the whole adjacent chamber and ample lateral buffer. If no small intercept exists, the analogous box contains no zero above the base up to height \(10c_1P_0\). An intervening zero would extend through the axis to a prohibited smaller intercept. The same statements hold below. All these chamber heights tend to infinity, since \(d(0)\ge\kappa^{-1}\log(1/B)\).

If \(E(0)\ge J_0(0)\), then \(d(0)=\kappa^{-1}\log(1/E(0))=O(\log(1/B))\). For \(A_0\) sufficiently large it is much smaller than \(\theta P_0\). Every realizing contact lies in the preceding graph system. A contact on the base sheet, including a diagonal one, has value at least \(cB^{-1/4}\gg P_0\), by its curvature bound and the graph Taylor formula. Every other competing sheet has its shortest chord in the plateau, while every chord to it has penalty at least \(G_1\). Consequently \[ d(0)=G_1\ell_{\min}, \tag{104}\] where \(\ell_{\min}\) is the shortest distance to an adjacent graph. Only adjacent graphs can be shortest, since a segment to a farther graph crosses an intervening zero.

Step 2: the opposite sheet gives the lower interaction bound. For a fixed unit vector \(n_0\), put \(w=q(1-p\cdot n_0)\ge0\). The identities for \(q\) and \(\nabla v\), together with \(q p=\sqrt2\nabla v\), give \[ (-\Delta+2)w=3qw-q^2D\ge qw\ge0. \tag{105}\] The last inequality uses \(D\le2(1-p\cdot n_0)\). The positive kernel of \(-\Delta+2\) is \[\mathcal K(x)=\int_0^\infty(4\pi t)^{-n/2} e^{-2t-|x|^2/(4t)}\,\,\mathrm dt.\] For \(r\ge1\) it is bounded above and below by positive constants times \(r^{-m/2}e^{-\sqrt2 r}\); the bounds follow by splitting this integral about its minimum at \(t=r/(2\sqrt2)\) and scaling the central interval by \(\sqrt r\). Convolution represents bounded solutions of this equation, by the maximum principle for their bounded homogeneous difference. In particular a nonnegative subset of its source gives a lower bound.

Take \(n_0=e_n\) and assume \(E\ge J_0\). On the opposite adjacent sheet realizing (104), the oriented normal is almost \(-e_n\), because the signs of a continuous function alternate between its regular zero graphs. On a tangent disk of radius \(c\sqrt{\ell_{\min}}\) and a fixed thin neighborhood of this disk, \(q w\) has a fixed positive lower bound: \(D<1/4\) at its zeros and uniform local derivative bounds apply. The distances to \(0\) are \(\ell_{\min}+O(1)\), and its volume is comparable to \(\ell_{\min}^{m/2}\). Thus (105) gives \[D(0)\ge w(0)\ge c e^{-\sqrt2\ell_{\min}}=cE(0).\] This proves the second inequality of (102), including its exact exponential rate. The remaining inequality requires an upper bound for the deficit of the base slope; a one-sided curvature error would be too large for that purpose.

Step 3: cancellation for two artificial half-space solutions. Write the base graph as \(x_n=h(y)\), with \(h(0)=0\) and \(\nabla h(0)=0\), and cut it off smoothly outside the preceding comparison region at scale \(P_0\). Its extension can be chosen with \[\|h\|_{C^2}=o(1),\qquad |\nabla h(y)|+|\nabla^2h(y)| \le C\sqrt{J_0(0)}e^{2\lambda|y|}.\] For an explicit bound independent of \(P_0\), take the cutoff equal to \(1\) on \(|y|\le P_0\) and supported on \(|y|\le2P_0\). The uncut graph has \(|\nabla h|\le1\), so \(|(y,h(y))|\le2|y|\) and \(|\nabla^2h(y)|\le C\sqrt{J_0(0)}e^{\lambda|y|}\). Radial integration from \(h(0)=\nabla h(0)=0\) gives \[|\nabla h(y)|\le C\lambda^{-1}\sqrt{J_0(0)}e^{\lambda|y|}, \qquad |h(y)|\le C\lambda^{-2}\sqrt{J_0(0)}e^{\lambda|y|}.\] Derivatives of the cutoff contribute factors \(P_0^{-1}\) and \(P_0^{-2}\), which are at most \(1\) for \(P_0\ge1\). Thus the extended graph obeys the displayed weighted estimate with a constant at most \(C(1+\lambda^{-1}+\lambda^{-2})\), fixed after \(\lambda\), and no factor growing with \(\log(1/B)\). Its unweighted \(C^2\) smallness follows separately from \(P_0^2B^{1/4}\to0\). Use this same extension \(h\) on both sides. In the coordinates \(x_n=h(y)\pm t\), \(t>0\), seek positive solutions \(v^*_\pm=g(t)+\varphi_\pm\) with zero boundary. Their equations are \[ \begin{split} L_g\varphi_\pm={}&|\nabla h|^2\partial_{tt}(g+\varphi_\pm) \mp2\nabla h\cdot\nabla_y\partial_t\varphi_\pm \mp\Delta h\,\partial_t(g+\varphi_\pm)\\ &-3g\varphi_\pm^2-\varphi_\pm^3. \end{split} \tag{106}\] Lemma 33 and the embedding \(W^{2,p_0}_{\rm uloc}\subset C^{1,\eta}\) make the right-hand side followed by the inverse a contraction in a small unweighted ball. The highest derivative terms have coefficients tending uniformly to zero; the nonlinear terms have Lipschitz constants tending to zero on that ball. The source terms from \(g\) tend to zero there. This proves existence and uniqueness of small zero-trace solutions; local elliptic bootstrapping gives smoothness. Weighted estimates in the same equation, with the small-coefficient terms absorbed, then give \[\|\varphi_\pm\|_{W^{2,p_0}(B_1(y,t)\cap\{t>0\})} \le C\sqrt{J_0(0)}e^{3\lambda|y|}.\] They are positive by \(C^1\) smallness near \(t=0\) and \(C^0\) smallness away from it. They are at most \(1\): a supremizing sequence with value above \(1\), translated in the interior and using uniform regularity, gives a positive interior maximum incompatible with \(\Delta v^*=v^{*3}-v^*\).

Put \(S=\varphi_++\varphi_-\) and \(T_*=\varphi_+-\varphi_-\). Adding (106) gives exactly \[\begin{split} L_gS={}&|\nabla h|^2(2g''+S_{tt}) -2\nabla h\cdot\nabla_y(T_*)_t-\Delta h\,(T_*)_t\\ &-3g(\varphi_+^2+\varphi_-^2) -(\varphi_+^3+\varphi_-^3). \end{split}\] The terms \(\mp\Delta h\,g'\) cancel. Every remaining source is \(O(J_0(0))\) in the weighted local \(L^{p_0}\) norm with exponent \(10\lambda\): the coefficient factors have weight \(2\lambda\), the perturbations have weight \(3\lambda\), and the cubic terms also have a uniform small bound on one factor. Taking \(10\lambda<b_0\) and using the inverse and boundary Sobolev embedding, we conclude \[ |\partial_t(\varphi_++\varphi_-)(0,0)|\le C J_0(0). \tag{107}\]

Thus the sum of the two artificial boundary slopes differs from the planar value by a curvature-square envelope. We next estimate the change in each slope caused by the actual adjacent sheet, if present.

Step 4: comparing an artificial solution with the true chamber. We prove, on the upper side with a small adjacent intercept, \[ |\partial_n(v-v^*_+)(0)| \le C\bigl(e^{-\sqrt2\ell}+e^{-cP_0}\bigr). \tag{108}\] If no small intercept is present, the right side is \(Ce^{-cP_0}\). The lower-side assertion compares \(-v\) with \(v^*_-\). All constants \(c>0\) below are fixed before choosing \(A_0\).

Middle-chamber deficit. First we establish a sharp bound at the middle of a chamber. Set \(t_1=x_n\) and \(t_2=\ell-n'\cdot x\). In the central lateral box, near \(t_1=t_2\), both \(1-v\) and \(1-v^*_+\) satisfy \[ 1-v,\ 1-v^*_+ \le C\bigl(e^{-\sqrt2 t_1}+e^{-\sqrt2 t_2}+e^{-cP_0}\bigr). \tag{109}\] Here and below \(v\) is positive in the chamber. For the proof, subtract the uniform \(o(1)\) graph errors from \(t_1\) and \(t_2\), and add the lateral face distances of a wider box. Call all these inward unit affine distances \(l_1,\ldots,l_k\). Their positive intersection is contained in the phase, by ordering. For either deficit \(u=1-v\) or \(u=1-v^*_+\), \[(-\Delta+2)u=3u^2-u^3.\] Sign-ball clearing, as proved with (41), makes \(u\) uniformly as small as desired where \(\min_jl_j\ge M\), for large fixed \(M\). Put \(a_0=\sqrt2\) and choose positive constants \(C_j\) with \(C_je^{-a_0M}=\eta_0\), where \(\eta_0>0\) is fixed small. On this inner polytope the function \[\mathcal W=\sum_{j=1}^k \left(C_je^{-a_0l_j}-kC_j^2e^{-2a_0l_j}\right)\] is positive and at most \(k\eta_0\). With \(r_j=C_je^{-a_0l_j}\), \[(-\Delta+2)\mathcal W=6k\sum_jr_j^2 \ge6\Bigl(\sum_jr_j\Bigr)^2\ge3\mathcal W^2.\] Choose \(\eta_0\) so \(k\eta_0<1/8\), then \(M\) so clearing gives \(u\le\eta_0/2\) on the boundary. On a boundary face \(\mathcal W\ge\eta_0-k\eta_0^2\ge\eta_0/2\). The difference equation has positive zeroth coefficient while \(u,\mathcal W\) are small, so the maximum principle gives \(u\le\mathcal W\). The central point is a distance comparable to \(P_0\) from the lateral faces; the \(o(1)\) shifts only change constants. This proves (109).

Potential comparison on the lower half of the chamber. Now use the domain between \(x_n=h(y)\) and the affine midpoint \(t_1=t_2\), with \(|y|_\infty<c_2P_0\). In the case without a small intercept, use instead its horizontal top at height \(c_2P_0\). The difference \(u_*=v-v^*_+\) solves \[(-\Delta+V_*)u_*=0, \qquad V_*=v^2+vv^*_++(v^*_+)^2-1.\] For some fixed \(\gamma_1>0\), \[ V_*(x)\ge V_g(t_1)-\eta(B)e^{-\gamma_1t_1},\qquad \eta(B)\longrightarrow0. \tag{110}\] We check both the fixed-depth and increasing-depth content of this statement. At each base point in the lateral box there is a tangent ball in the positive phase whose radius tends uniformly to infinity. It may be chosen much smaller than \(c_1P_0\), the adjacent height, and \(B^{-1/4}\): the base Hessian bound keeps the ball above the base, and the chamber and lateral buffers exclude any other zero from it. For a concrete uniform choice, put \(H_B=\ell\) when the small adjacent intercept is present and \(H_B=c_1P_0\) otherwise, and take \[r_B=\bigl[\min\{c_1P_0,H_B,B^{-1/4}\}\bigr]^{1/2}.\] All three quantities in the minimum tend to infinity, so \(r_B\) is smaller than each by a factor tending to zero. The adjacent height over the larger box is bounded below by a fixed positive fraction of \(H_B\), and every base in the smaller box has a lateral margin comparable to \(c_1P_0\). Finally \(r_B B^{1/4}\to0\), so the Hessian bound excludes a second contact of the tangent ball with the base. The radius therefore works uniformly, including at bases on a lateral face of the smaller box. Lemma 25, or its explicit inequality (87), gives \(v\to g(t_1)\) on every fixed depth interval. Indeed, along the inward normal at a base point, that inequality and \(|\nabla U|\le1\) give \(s-C(1+s)/r\le U\le s\) for fixed \(s\ge0\), where \(r\) is the radius of the tangent phase ball. These radii tend uniformly to infinity. To pass from normal depth to the vertical coordinate \(t_1\), use the uniformly vanishing slope and height of the base graph and the same Lipschitz bound for \(U\). Thus the convergence is uniform over all bases in the buffered comparison box; no two-sided estimate for \(H\) is used. The artificial solution has the same convergence by construction, and \(h=o(1)\) on this box. At increasing depths up to the midpoint there is an explicit phase-ball radius independent of \(\theta\). Let \(\eta_B=o(1)\) bound the signed affine errors of both sheets in the larger box \(|y|_\infty\le10c_1P_0\). For a point in the closure of the smaller lateral comparison box, put \(s=t_1\ge1\). We have \(t_2\ge s\) and \(s\le\theta P_0/4\); the latter follows from \(\ell\le\theta P_0/10\) and (103). Set \(c_{\rm ball}=\min(1/8,c'_0)\). On \(B_{c_{\rm ball}s}(x)\) both unit affine distances are at least \(7s/8>\eta_B\) for small \(B\). Its horizontal displacement is at most \(c'_0\theta P_0/4=c_1P_0/4\), so it remains inside the larger box even when its center is on a lateral face of the smaller box. It therefore lies in the true positive chamber. It also lies above the artificial graph. Without a small intercept, \(s\le c_2P_0\), and the same ball stays below the empty-chamber height \(10c_1P_0\). Sign-ball clearing consequently gives \(1-v+1-v^*_+\le Ce^{-c_3t_1}\) with \(c_3>0\) depending only on the fixed \(c_{\rm ball}\) and dimension, before \(\theta\) is chosen. Take \(0<\gamma_1<\min(c_3,\sqrt2)\). Splitting at a fixed large depth and then using fixed-depth convergence proves (110). This reasoning includes the domain with the artificial upper face.

The barrier at rate \(\sqrt2\). We have bounded the error in the potential by a decaying function with coefficient \(\eta(B)\to0\). We now absorb that error into the comparison operator. This is the step that preserves the exact rate. Choose a fixed small \(\theta_0>0\); all \(t_1\) in the domain are at least \(-\theta_0\) for small \(B\). Put \(a_0=\sqrt2\) and \[Y(t)=g'(t)\left[1+\int_{-\theta_0}^t g'(s)^{-2}\,\,\mathrm ds\right], \qquad k'(t)=-Y(t)^{-2}\int_{-\theta_0}^tY(s)^2e^{-\gamma_1s}\,\,\mathrm ds.\] Then \(Y>0\), \((-\partial_t^2+V_g)Y=0\), and \(Y(t)\asymp e^{a_0t}\) as \(t\to\infty\). Consequently \(|k'(t)|\le Ce^{-\gamma_1t}\) and its integral is finite. Choose \(k(-\theta_0)\) so \(1\le k\le C\), and set \(\phi=e^{-a_0t}Yk\). Direct differentiation gives \[ \begin{gathered} 0<c\le\phi\le C,\qquad |\phi'|\le Ce^{-\gamma_1t},\\ -\phi''-2a_0\phi'+(V_g-2)\phi =e^{-a_0t}Y(t)e^{-\gamma_1t} \ge c e^{-\gamma_1t}. \end{gathered} \tag{111}\] These bounds also hold on the fixed compact initial interval. If \(c'=n'\cdot e_n\), then \[(-\Delta+V_*)\bigl[e^{-a_0t_2}\phi(t_1)\bigr] =e^{-a_0t_2} \bigl[-\phi''-2a_0c'\phi'+(V_*-2)\phi\bigr].\] By (111) the bracket is nonnegative when \(|1-c'|\) is a sufficiently small fixed number and \(B\) is sufficiently small. Its errors are bounded by \(C(|1-c'|+\eta(B))e^{-\gamma_1t_1}\). This is the reason the exact exponential \(e^{-\sqrt2t_2}\) survives. The small geometric and potential errors have been absorbed in the operator inequality before comparison. They are not additive errors in the interaction estimate and are never divided by its exponentially small scale.

There are also coarse barriers for the remaining faces. Take \(v_0\) from Lemma 23 with initial endpoint \(-\theta_0\). It is bounded above and below by positive constants and satisfies \((-\partial_t^2+V_g)v_0=1\). For any inward affine unit face distance \(l\), the function \(v_0(t_1)e^{-c_4l}\) is a supersolution of \(-\Delta+V_*\) when \(c_4>0\) is sufficiently small and \(B\) is sufficiently small. Indeed the new terms are bounded by \(C(c_4+c_4^2+\eta(B))\), whereas the undisturbed positive term is \(1\).

The difference is zero at the bottom. Multiples of these coarse barriers dominate its absolute value at the lateral faces. At the midpoint, (109) and \(t_1=t_2\) bound it by a multiple of \(e^{-a_0t_2}\phi(t_1)\) plus \(e^{-cP_0}v_0(t_1)\). The latter is itself a supersolution after multiplication by the constant \(e^{-cP_0}\). In the case with an artificial top face use its coarse barrier instead. Comparison is legitimate despite the possible sign of \(V_*\): division by \(v_0\) gives a strictly positive zeroth coefficient, by (110). The resulting bound on a fixed neighborhood of \(0\) is \(C(e^{-a_0\ell}+e^{-cP_0})\), or \(Ce^{-cP_0}\) without an adjacent small intercept. Boundary gradient estimates for the homogeneous equation on the uniformly smooth graph, with zero boundary value, prove (108).

Step 5: choosing the constants and recovering the base defect. The artificial cancellation and the true-chamber comparison are now proved separately. To combine them uniformly, use the following order for the constants. First fix the decay and comparison constants for \(g\), the half-space inverse, and the elementary phase-ball clearing. The box construction above has uniformly controlled aspect ratios when both fractions are proportional to \(\theta\), so choose \(\gamma_1\) and then \(\phi,v_0\) using those uniform constants. Next take \(\theta\) small enough for the plateau condition and the drift error in (111); choose \(c_1,c_2\) as specified. The resulting coarse exponent \(c>0\) is fixed. Now choose \(A_0\) large enough for the realizing-contact argument and \(cA_0>1\). Then choose \(\lambda\) small enough for \(10\lambda A_0<1/2\) and \(10\lambda<b_0\). Finally decrease the upper threshold for \(B\) to meet all small-error conditions. This order is independent of \(\delta\). In particular \(e^{-cP_0}\le B\) with uniform constants.

At \(0\), the two unsigned true derivatives in the inward height coordinates are both \(\partial_nv(0)\). Equations (107) and (108) show \[|2\partial_nv(0)-2g'(0)| \le C\bigl(J_0(0)+E(0)+e^{-cP_0}\bigr)\le CB,\] because every adjacent gap used satisfies \(d(0)\le G_1\ell\). Since \(D(0)=1-2\partial_nv(0)^2\) and \(g'(0)=1/\sqrt2\), this proves the first inequality of (102) and completes the proposition. ◻

Proof. At a good zero, \(|\nabla v|\ge\sqrt{3/8}\). Uniform second derivative bounds for \(v\) give a fixed ball on which its derivative along the initial normal is bounded below. All zeros in a smaller ball therefore form one graph, with uniformly bounded slope and derivatives. A separated selection of such smaller balls has bounded cardinality inside a unit ball, by ordinary ambient packing, and covers its good zeros. Summing their graph areas gives the claim. Larger balls are covered by \(C(1+r)^n\) unit balls. The last assertions are local Harnack and the differentiated bounds in Lemma 8, applied on a fixed regular graph contained in the good set. ◻

Proof. Multiplying the first inequality of (102) by \(D\) and the second by \(E\), and absorbing \(DJ_0,EJ_0\) by Young’s inequality, gives \(D^2+E^2\le C(J_0^2+DE)\). We estimate \(J_0^2\) without replacing a supremum by an unjustified point value integral. Let \(w\) be good and \(\mathop{\mathrm{dist}}(w,X)>2\). On its fixed regular graph, for \(w'\) within intrinsic distance \(\mu\), Lemma 34 gives \[a(w)^4\le C a(w')^4+C\mu^4D(w)^2.\] Average on that small graph disk, and use Harnack and a fixed-size graph disk to bound \(D(w)^2\) by the integral of \(D^2\) in \(B_1(w)\). For every sufficiently small fixed \(\mu>0\) this yields \[a(w)^4\le C_\mu\int_{B_1(w)\cap(\Sigma\setminus X)}a^4\,\,\mathrm dS +C\mu^4\int_{B_1(w)\cap(\Sigma\setminus X)}D^2\,\,\mathrm dS.\] The coefficient in front of the last integral is independent of \(\mu\). Values at \(w\) within distance \(2\) of \(X\), including bad points where \(\bar a=1\), are bounded by a constant and can instead be charged to some point of \(\mathcal X\) within distance \(3\). Since bounded shifts change \(e^{-2\lambda|z-w|}\) by only a fixed factor, taking the supremum over \(w\) gives \[J_0(z)^2\le C\int e^{-2\lambda|z-w|} \left(C_\mu a^4\,\,\mathrm dS+C\mu^4D^2\,\,\mathrm dS +\,\mathrm d\#_{\mathcal X}\right)(w).\] The surface measures here are restricted to \(\Sigma\setminus X\). By Lemma 34 and \(\chi_R(z)\le e^{|z-w|/R}\chi_R(w)\), for \(R\ge2/\lambda\), \[\int_{\Sigma\setminus X} e^{-2\lambda|z-w|}\chi_R(z)^2\,\,\mathrm dS_z \le C_\lambda\chi_R(w)^2.\] The constant is independent of the good threshold, by summation over unit annuli and their polynomial area bound. Tonelli now bounds the integral of \(J_0^2\chi_R^2\). Choose \(\mu\) small enough to absorb the resulting \(D^2\) term, after the constant in (102) and \(\lambda\) have been fixed. This proves (112). All integrals are finite: \(D,E,a\) are bounded on good zeros, the area has polynomial growth, and the separated net has polynomial ambient growth. ◻

Proof. We establish graph continuation on a polynomial scale before using it on any logarithmic scale. Put \[\kappa_N=CN^{-\omega/2},\qquad r=N^{\omega/8}.\] At every zero under consideration, \(|A|\le\kappa_N\) and \(|\nabla v|\) has a fixed positive lower bound. The fractional margin contains all balls of radius \(100r\) used below once \(N\) is large. Starting at any such zero, lift a straight radial segment in its tangent plane by tangent projection. As long as the lift has length at most \(10r\), its normal changes by at most \(10\kappa_Nr=o(1)\). The differential of projection therefore remains invertible, and the lifted length is at most twice the projected length. In particular every segment of projected length at most \(r\) continues to its endpoint: a terminating sequence stays in the larger ball, has a regular zero as its limit, and local inversion continues it. The lifting equation depends smoothly on the segment endpoint. Its endpoints thus form one smooth graph over the full tangent disk; local uniqueness identifies it with the original zero piece. Starting with a larger fixed multiple of \(r\) supplies the stated buffer. Taylor’s formula on these graphs gives the affine error \[C\kappa_Nr^2=O(N^{-\omega/4}).\]

We next put nearby pieces over the same plane. Two such graph disks whose base points are at distance \(b=o(r)\) have unoriented normal angle at most \[ C\left(\frac b r+\kappa_Nr\right). \tag{113}\] Indeed, if the angle were larger, a line in one tangent plane in the direction of maximal tilt would make its signed distance from the other tangent plane take opposite signs, with magnitude greater than both affine errors, inside a fixed smaller disk. Points of that graph project into the interior of the other disk. The signed graph height, measured in the second disk’s normal direction, then vanishes by the intermediate value theorem. At this intersection the two normals would still differ by more than their variations \(C\kappa_Nr\), contradicting the unique tangent plane of the regular level set. This argument also excludes differently tilted pieces of a single connected component. Taking \(b=O(\log N)\) proves the claimed common-plane slope bound. Over a common smaller disk the graphs either agree or are disjoint, and so are ordered.

For completeness, no new sheet can enter unnoticed through the side of a logarithmic-power box. A zero \(w\) with lateral coordinate \(|y|\le(\log N)^k\) and height at most \(C_0\log N\) first has \(b=O((1+\log N)^{\max(k,1)})\) in (113). This already gives a slope tending to zero polynomially and permits its graph to continue to the normal axis through \(z\). Its height changes on that continuation by at most \[C\left(\frac{(1+\log N)^{\max(k,1)}}r+\kappa_Nr\right) (1+\log N)^k=o(1).\] Its intercept is therefore within \((C_0+1)\log N\) of \(z\) for large \(N\). Applying (113) again at this intercept gives the refined slope bound already stated. Enlarging the fixed depth constant in a buffered description therefore includes all pieces in the smaller box. There are only finitely many in each closed inner cylinder: otherwise zeros on distinct graphs accumulate at a regular zero, whose uniform local graph neighborhood contains only one zero piece. This is local finiteness, and imposes no global bound on the number of sheets.

The same estimates give the required additive accuracy. On a disk of radius \(L=(\log N)^k\), the error from a graph’s own tangent plane is at most \(C\kappa_NL^2\). If \(\vartheta_N=C(\log N/r+\kappa_Nr)\) denotes the common-plane slope bound, changing from its tangent plane to the common horizontal plane costs at most \(C\vartheta_NL+C\kappa_NL^2\) in height. Both quantities are bounded by a negative power of \(N\) times a fixed logarithmic power. They tend to zero as numbers, rather than only relative to the gap.

To check projections, write one graph as \(x_n=f(y)\). For a fixed point \(x\), half its squared distance to \((y,f(y))\) has Hessian \[\mathrm{Id}+\nabla f\otimes\nabla f-(x_n-f)D^2f.\] On a buffered disk relevant to logarithmic depth this is positive definite, with eigenvalues \(1+o(1)\): the slope tends to zero and \(|x_n-f||D^2f|=o(1)\). The same bound holds out to a fixed fraction of the polynomial disk, since its height variation is at most \(C\vartheta_Nr+C\kappa_Nr^2=O(\log N)\). A minimizing point cannot occur at that outer boundary, whose horizontal distance is polynomially large, whereas the vertical competitor has distance \(O(\log N)\). Strict convexity gives a unique foot, and the implicit function theorem gives its smooth dependence on \(x\). The stationary equation bounds the horizontal displacement of the foot by \(C\vartheta_N\log N\) and the difference between true and vertical distances by \(C\vartheta_N^2\log N\), after harmless enlargement of the common slope bound. These are the asserted additive errors. This projection is to a specified local graph, not to an entire connected component. Every foot that can compete for the nearest zero lies within logarithmic distance, and hence belongs to the buffered family by the preceding completeness argument.

To prove separation, set \(r_0=c_0\log N\) with \(c_0>0\) sufficiently small. Starting at any participating zero, the exponential bound in (36) gives \(D\le CN^{-\omega/2}\) throughout \(B_{4r_0}\). Consequently \(|T|\le CN^{-\omega/4}\) there. Integration shows that \(p\) remains uniformly close to the initial unit normal, so its component in that direction is positive. All zeros in a smaller cylinder then form a single graph. In conjunction with the common graph description, this excludes a distinct sheet at distance less than a smaller fixed multiple of \(r_0\).

Finally assume a distinct sheet lies within \(C_0\log N\) of \(z\). Its nearest foot is interior to the polynomial graph disk, and the shortest chord is normal to that graph. The common normal estimate puts its angle at \(z\) in the plateau of \(\Gamma\), for large \(N\). It supplies a penalized competitor of size \(O(\log N)\). Every better competitor has chord length \(O(\log N)\) because \(\Gamma\ge1\), so lies in the same graph system. A chord of length \(l\) to the base graph has normal displacement at most \(C\kappa_N l^2\). Since \(\Gamma(s)\ge1/s\), its penalized length is at least \(c\kappa_N^{-1}\gg\log N\), also in a diagonal limit. For every other graph all chords cost at least \(G_1\) times their length, and its shortest chord attains this factor. Only adjacent graphs can minimize, since a chord to a farther one crosses an intervening zero first. This proves the exact identity for \(d\). All graph identifications and projections on overlapping inner windows agree by their local uniqueness; all construction scales fit within the original fractional margin. ◻

Proof. For (114) let \(\ell=\ell_{\min}\) and fix \(n_0=\nu(z)\) in (105). On the column \(z+s\nu(z)\), \(|s|\le P\), (101) and local Harnack imply \[D=2(1-p\cdot n_0)-|p-n_0|^2, \qquad qD\ge(2-o(1))w.\] In fact \(|p-n_0|^2=O_P(D(z)^2)\) whereas \(D\) is comparable to \(D(z)\) there, so the last error is uniform and relative. At the opposite foot use a tangent disk of radius \(\ell^{2/3}\) and bounded depth \(|t|\le Q\) in its affine normal coordinates. The true orientations are opposed, and all profile and affine errors on this patch tend uniformly to zero by Lemma 36 and (101) at its bases. Thus the exact source on the right of (105) satisfies \[3qw-q^2D\ge(6-o(1))G(t)^2,\qquad |t|\le Q.\] The constant is \(6\), since \(w\to2G\) there. For fixed \(P,Q\) the comparison is relative, as \(G\) has a positive minimum on the compact depth interval.

Integrating the heat-kernel formula for \(\mathcal K\) in its horizontal variables gives exactly \[\int_{\mathbb R^m}\mathcal K(y,h)\,\,\mathrm dy =\frac1{2\sqrt2}e^{-\sqrt2|h|}.\] Truncating this integral to \(|y|\le\ell^{2/3}\) changes it by a relative \(o(1)\) for \(h=\ell+O(P+Q)\): the excess distance outside that disk is at least \(c\ell^{1/3}\), and the kernel bounds used above control the remaining polynomial volume factor. Polynomial-small graph and angular errors contribute only \(o(1)\) additive errors to distances on this disk, so do not change the exponential factor by more than \(1+o(1)\). It follows, with signs depending on the side but irrelevant by symmetry of \(G\), that \[\begin{split} \frac{I_P(z)}{e^{-\sqrt2\ell}} &\ge \frac{2-o(1)}\sigma\frac{6-o(1)}{2\sqrt2} \left(\int_{-P}^{P}e^{\sqrt2s}G(s)^2\,\,\mathrm ds\right) \left(\int_{-Q}^{Q}e^{\sqrt2t}G(t)^2\,\,\mathrm dt\right). \end{split}\] Choose \(P,Q\) large and fixed, then \(N\) large. The elementary substitution \(r=e^{\sqrt2t}\) gives \[\int_\mathbb Re^{\sqrt2t}G(t)^2\,\,\mathrm dt=2\sigma.\] The limiting coefficient is therefore \[\frac2\sigma\frac6{2\sqrt2}(2\sigma)^2=32,\] which proves (114).

The complementary estimate concerns the value of the phase between the same sheets. Here the coefficient comes from the one-dimensional tail \(1-g(s)\sim2e^{-\sqrt2s}\) and from \(q\le2(1-v)\) in a positive phase. The lateral boundary must be placed farther away than the logarithmic sheet separation so that its contribution is negligible relative to the principal tails.

For (115), change the sign of \(v\) if necessary so the phase between the sheets is positive. Separation and \(|s_i-s_j|\le C_1\) imply \(\min(s_i,s_j)\to\infty\), so the point \(x\) belongs to the fixed-depth inner polytope used below once \(N\) is sufficiently large. Use the two tangent planes at the perpendicular feet from \(x\), together with lateral faces at scale \((\log N)^2\). Lemma 36 provides these graphs with polynomial buffers, no intervening zeros, and affine errors \(o(1)\). Subtract these errors from the two face distances, producing an inner polytope as in the proof of (109). Its distances at \(x\) are \(s_i+o(1)\) and \(s_j+o(1)\).

In that proof the two principal coefficients \(C_j\) can now be chosen equal to \(2+\eta\), with arbitrarily small fixed \(\eta>0\). Indeed at every point of their inner boundaries \(l_j=M\), including intersections with lateral faces, the true perpendicular projection is unique and remains in the polynomially buffered clean window. The signed distance to its sheet is \(M+o(1)\): the affine error on the entire \((\log N)^2\) box is polynomial-small, including its gradient, and the squared-distance Hessian in Lemma 36 is positive definite. Fixed-depth profile convergence at these feet is uniform, so the solution converges there to \(g(M+o(1))\). Since \(1-g(M)=2e^{-\sqrt2M}+O(e^{-2\sqrt2M})\), choose \(M\) large so that \[(2+\eta)e^{-\sqrt2M} -k(2+\eta)^2e^{-2\sqrt2M}>1-g(M)\] with fixed positive slack, and then \(N\) large enough to retain it. Keep the coarse fixed coefficients on the lateral faces. Each summand in the barrier is nonnegative; thus intersections of faces cause no difficulty in the boundary comparison. The same algebra \((-\Delta+2)\mathcal W\ge3\mathcal W^2\) applies with these unequal coefficients. At \(x\), the lateral contribution is \(O(e^{-c(\log N)^2})\), hence negligible relative to either of the two exponentials, since \(s_i,s_j\le C_0\log N\). We obtain \[1-v(x)\le(2+2\eta) \left(e^{-\sqrt2s_i}+e^{-\sqrt2s_j}\right).\] Finally \(q=(1-v)(1+v)\le2(1-v)\); take \(\eta\) sufficiently small in terms of \(\varepsilon\) to prove (115). ◻

Proof. The strict gradient gain lets Simons’ identity and surface stability control one another. We derive that gain first, apply it away from \(X\), and then charge the omitted collar to \(N_R\).

Proof. If \(X\) is empty, apply (100) to \(\chi f(d)\) and use Young’s inequality. Suppose henceforth that \(X\) is nonempty. We construct the test, reserve positive mass for its transition region, and estimate its derivatives. The \(f'\) energy and the spatial-weight cost remain on the right side of (123). After estimating those terms and using the mass reserve, only the two partition terms remain to be canceled by the variance identity.

Proof. There are four ranges. A plateau protects the collar; a very slow power decay makes the multiplier flux on that plateau small; a faster power decay makes \(t^3f^2\) small; and the final exponential tail matches \(E=e^{-\kappa d}\). Take \(t_s\ge t_*\) and use the continuous piecewise formula \[f(t)= \begin{cases} 1,&t\le t_s,\\ (t/t_s)^{-\mu},&t_s<t<t_p,\\ f(t_p)(t/t_p)^{-k},&t_p<t<t_e,\\ f(t_e)e^{-\kappa(t-t_e)/2},&t\ge t_e. \end{cases}\] Here \(0<\mu\ll1\), \(t_p>t_s\), and \(t_p<t_0<t_e\). Corners cause no problem for locally Lipschitz tests or the multiplier identity. Set \(F_0=\exp(C/(\gamma t_{\min}^{\gamma}))<1.001\). For \(s\ge t\), \[\frac{p_*(s)}{p_*(t)} \le F_0(s/t)^{1+\zeta_0}.\]

Proof. For \(t>L\), \[b'=(b/t)^\vartheta,\qquad t^{-\vartheta}\,\mathrm dt=b^{-\vartheta}\,\mathrm db.\] Since \(c\tau>1\) for large \(R\), the part \(t\le L\) contributes nothing to \(\mathcal F_\tau\). Thus \[\mathcal F_\tau =c\int_{b(t)}^\infty u^{-\vartheta}f_*'(c\tau-u)\,\mathrm du.\] The total mass of the measure \(f_*'(c\tau-u)\,\mathrm du\) in this integral is \(f_*(c\tau-b(t))\). If \(b(t)\le c\tau+1\), this mass is concentrated at \(u=c\tau+O(1)\) with an exponentially decaying right tail; if \(b(t)>c\tau+1\), it is concentrated at \(u=b(t)+O(1)\) with the same property. Its support is contained in \(u\ge c\tau-1\) in the first case. Comparing \(u^{-\vartheta}\) with \((c\tau\vee b(t))^{-\vartheta}\) proves the first estimate uniformly; the same comparison with \(u^{-\vartheta-1}\) proves the bound for \(J_1\).

The logarithmic slope satisfies \[ 0\le-\partial_t\log W_*=\vartheta/t+\lambda b'\le q_*+\vartheta/t_{\min},\qquad 0\le\partial_\tau\log W_*\le cq_*. \tag{140}\] Integration of the resulting exponential lower bound for \(W_*(\tau,t+s)/W_*(\tau,t)\) gives the lower ratio in (138). The same argument starting at the current \(t\) gives \(\mathcal F/W_*\ge1/(q_*+\vartheta/t)\). In particular, on any region where \(d\ge d_{\min}(R)\to\infty\), the ratio is at least \(1/(2.2+o_R(1))\) uniformly. The relation \(t\le C(L+b(t))\), splitting the integrals at \(C(L+\tau)\), and the exponential tail of \(f_*\) give the two upper bounds. In the exponentially decaying region one can also use \[\mathcal F/W_*\le C(t/b(t))^\vartheta\le C \quad\text{if }t>C(L+\tau).\] This observation will allow polynomial factors in \(d\) to be discarded far beyond the switching region.

For completeness the derivatives responsible for the gradient square are \[\mathcal F_t=-W_*,\quad \mathcal F_{tt}=W_*(\vartheta/t+\lambda b'),\quad \mathcal F_{\tau t}=-cW_*\lambda,\quad \mathcal F_{\tau\tau} =c^2\bigl(b^{-\vartheta}f_*'(c\tau-b)-\vartheta J_1\bigr).\] Since \(b^{-\vartheta}f_*=W_*/b'\), their quadratic expression in \((\nabla_\Sigma\tau,\nabla_\Sigma d)\) is exactly the last two terms of (139), minus \(c^2\vartheta J_1|\nabla_\Sigma\tau|^2\). Near \(t=L\) we have \(f_*'=0\); all terms involving \(\lambda/b'\) therefore vanish on a neighborhood of this apparent singularity. The reach is locally semiconcave, with nonpositive singular Laplacian part. Multiplication by \(\mathcal F_t=-W_*<0\) changes this part to a nonnegative measure. The smooth calculation for its almost-everywhere density consequently gives (139); one can equivalently obtain it by local concave smoothing and passage to the distributional limit. ◻

Proof. Take a uniformly separated net on \(Z\) in an enlarged shell. Outside a unit neighborhood of \(X\), each such center has a regular surface disk of fixed radius on which \(D\ge c s^{-\alpha}\), by (36). Its contribution to \(\int D^2\) is at least \(c s^{-2\alpha}\). Bounded overlap and (116) bound the number of these centers by \(C K(s)s^{m-4+2\alpha}\). Centers near \(X\) obey the stronger bad-net bound from the same estimate. Enlargement by \(Cs^{\beta_0}\) costs at most \(Cs^{n\beta_0}\), using unit ambient volume and unit good-area bounds.

For (141) use \(K(2^jR)\le C2^jK(R)\) and \(-2+2\alpha+n\beta_0=-1.85\). After division by \(\mathcal A\), its sum is bounded by \[C R^{-1.85}(1+\log R)^C \sum_{j\ge0}2^{-.85j}(1+j)^C=o(1).\] Here \((L+\log R+j)/L\le C(1+j)\) since \(L\ge c_0\sqrt R\). For (142), the elementary inequality \(\sqrt{a+b}\le\sqrt a+\sqrt b\) gives instead \[\begin{align*} \frac1{\mathcal A}\sum_{j\ge0} C K(s_j)e^{C\sqrt{\log(K(s_j)+1)}}s_j^{-2} (L+\log s_j)^{3-\vartheta} &\le C\frac{L^2}{R^2} e^{C\sqrt{\log(K(R)+1)}} \sum_{j\ge0}2^{-j}e^{C\sqrt{j+1}}(1+j)^C\\ &\le C\frac{e^{C\sqrt{\log(K(R)+1)}}}{K(R)} \longrightarrow0. \end{align*}\] The last limit uses only \(K(R)\to\infty\).

To prove (143), fix a sufficiently small \(\epsilon>0\). At fixed \(L\), \(b(t)\ge(1-\epsilon)t-C_{\epsilon,L}\) for all \(t>0\). Thus, on the good set, \[W_*\le C_Rs^{cq_*}E^p,\qquad p=\frac{q_*(1-\epsilon)}{\kappa}<2.\] Hölder’s inequality, (116), and the good-area bound give \(\int_{\rho\asymp s}E^p\le C K(s)s^{m-2p}\). Using \(K(s)\le Cs\) proves (143) with \[\epsilon'=q_*\left(\frac{2(1-\epsilon)}{\kappa}-c\right)-1>0.\] For example the limiting value at \(\epsilon=0\), \(\kappa=\sqrt2\) is greater than \(.197\), so the fixed choices above leave strict room. ◻

Proof. Integrate (139) against \(\phi m_c^2\) and move the surface Laplacian onto this test. The preceding error estimates justify this operation and control all derivatives of \(m_H,m_B\). The mean-curvature terms in the radial identities are at most \[C\sup_{\rho\ge R}\frac{L+\tau}{\rho} \int\phi m_c^2W_*|H|=o_R(1)\mathcal L\] by (138), (137), and \(|H|\le .355I\). The term involving \(J_1\) loses only \(o_R(1)\mathcal F_\tau/\rho^2\).

We spell out the source accounting in \(W_*(\Delta d)_{\rm upper}\). On a nonplateau branch where \(ad\le a_0\), (91) and \(|\nabla d|\ge c_\Gamma>0\) show that the sum of the \(a^2d\) term and its gradient error is bounded by \(.008|\nabla d|^2/d\), on first choosing \(a_0\) small and then the uniform lower reach large. Where \(ad>a_0\), (90)–(91) instead bound the source by \[C W_*a^4d^3e^{C\sqrt{\log(K(d)+1)}}.\] For \(d\le M_0(L+\tau)\ll\rho\) its shell integral against \(\phi\) is bounded by (142), using \(W_*\le d^{-\vartheta}\) and (116). Indeed, on \(\rho\asymp s\), the reach power contributes at most \(C(L+\log s)^{3-\vartheta}\), the subpolynomial factor is bounded using \(K(d)\le K(Cs)\), and the product of \(\phi\) with the fourth-order shell bound has size \(CK(s)s^{-2}\). The remaining tail was treated above. The gradient error on plateau branches is smaller than the same \(.008\) allowance.

The local mean curvature on a plateau costs \(G_1|H(z)|\). For its positive foot term use (93). On the truncated range the source and receiving radii have ratio \(1+o_R(1)\). Since \(d(y)\le d(z)\) and \(|\partial_\tau\log W_*|\le cq_*\), \[W_*(z)\le(1+o_R(1))W_*(y).\] If the receiving point lies in the rough collar, transfer first and integrate over that receiving set. In fact (93) and the uniformly diverging source reach imply \(D(y)\to0\), so the foot is good for the fixed threshold. The transfer Jacobian is bounded, \(W_*(z)\le d(z)^{-\vartheta}\le C\), \(H_{N_y}(y)_+\le CD(y)\le C\), and \(\phi(z)\le(1+o_R(1))\phi(y)\). Positive incoming multiplicity is at most one. Thus the charge is bounded by \[C\int_{\substack{y\in\Sigma\setminus X\\ \mathop{\mathrm{dist}}(y,Z)\le4\rho(y)^{\beta_0}}}\phi(y)\,\mathrm dS_y.\] Its shell bound is \(CK(s)s^{-2+2\alpha+n\beta_0}\), smaller than (141). No enlargement of the receiving collar by \(d(z)\) is made: even when \(d(z)\) is of order \(L\), the area formula counts the receiving set itself. Otherwise the point is on the good surface. For \(a(y)d(z)>a_1\), use \(H_{N_y}(y)_+\le C/d(z)\le C_{a_1}a(y)^4d(z)^3\), the uniformly bounded area distortion, and (116); this is another error of type (142). The reach power here remains the source value \(d(z)^3\): the truncated source and receiving points lie in comparable shells, and the combined factor \(W_*(z)d(z)^3\) is at most \(C(L+\log s)^{3-\vartheta}\) after transfer. For \(a(y)d(z)\le a_1\), choose \(a_1\) and \(\delta_2\) small so that the area distortion is at most \(1.01\). There is exactly the incoming multiplicity bound of (93), namely one.

Spatial cutoff transfer uses, with fixed small \(\epsilon_c>0\), \[m_B(z)^2\le(1+\epsilon_c)m_B(y)^2+ C_{\epsilon_c}(d/B)^2\mathbf1_{\{\rho\asymp B\}}.\] The latter term tends to zero by (143), since the truncated reach is at most \(C(L+\log B)\) on this shell. The radial factors \(\phi\) have ratio \(1+o_R(1)\) as well. Consequently the sum of all local and incoming curvature charges is at most \[(1+o_R(1))G_1(.355)\bigl(1+1.01(1+\epsilon_c)\bigr)\mathcal L <.73\mathcal L .\] This leaves room for the radial mean-curvature terms already listed.

It remains to give the constant mass on the left. If \(d\ge .8\tau\), then \(b(d)\le d\) and (138) gives \[(4-o_R(1))\mathcal F_\tau/W_* \ge 4(.87)(.8/.87)^{.02}-o_R(1)>3.44 .\] The nonnegative \(2T_r\mathcal F_\tau\) term may be discarded. On the exceptional set \(d<.8\tau\), \(E^2\ge s^{-1.6\kappa}\) on a shell \(\rho\asymp s\). Thus the omitted weighted mass \(\int\phi W_*/\rho^2\) is at most \(C K(s)s^{-4+1.6\kappa}\) by (116). Its dyadic sum is negligible, since \(1.6\kappa<3\) and \(K(s)\le C(s/R)K(R)\). The gradient coefficient left from \(\vartheta=.02\) is \(.012\). The second radial identity gives \(m(m-2)=24\), proving (144). ◻

Proof. Choose \(0<\omega<\min(\alpha,\beta_0)\), put \(N=\rho(z)\), and consider \(B_{2N^\omega}(z)\). Since \(\rho\) is \(1\)-Lipschitz, its values there are \(N+O(N^\omega)\). For large \(R\) this ball avoids \(Z\), because \(2N^\omega<N^{\beta_0}/3\). Every zero in it therefore satisfies \[D<\rho^{-\alpha}\le CN^{-\alpha}\le N^{-\omega}.\] Lemma 36 applies on a fixed fractional interior, decreasing \(\omega\) if necessary to leave a margin. Every fixed logarithmic-power disk and every fixed normal interval fit inside that interior. Its errors are bounded by a positive inverse power of \(N\) times a fixed logarithmic power, so their supremum for \(N\ge R\) tends to zero. The sharp analytic estimates (114)–(115) apply with these same premises after \(P\) and the depth constants are fixed.

We verify the extra conclusions used to count bands. In a frame over one of the common tangent planes, unsigned sheet distances differ additively by \(o_R(1)\) from distances to the ordered graph intercepts. Successive intercepts differ by at least \(c_0\log\rho\). Among ordered points on a line, two whose distances are within a fixed amount of the least distance must bracket the point and be adjacent; a third is farther by a diverging amount. The same assertions therefore hold for the true sheets.

Now fix a base and its positive normal ray, putting the base at height zero. The first adjacent graph on that side has intercept \(l=\ell_{ij}+o_R(1)\). Before reaching it, the graph estimates and smooth distance gradients give \[s_j=l-s_i+o_R(1),\qquad\frac{\,\mathrm ds_j}{\,\mathrm ds_i}=-1+o_R(1).\] These are the asserted sum and derivative formulas. The derivative of \((s_i-s_j)/2\) is positive, hence a fixed-width band is traversed at most once. A farther graph cannot be the partner before the first is reached, because the first intervenes and is closer. After crossing that first graph, it remains nearer than the base graph on the relevant logarithmic ray. Thus the base cannot meet a second band on this side. The negative side is identical. Finally, all descriptions concern the actual zero pieces and their unique projections, so they agree on overlaps. ◻

Proof. We separate the terms whose coefficients tend to their planar values from the normal derivative of the interpolation. This leaves a one-gap calculation and the count that allows a positive charge only once per base.

Proof. The energy in \(B_R\) is at least \(c K(R)R^m\) by the center choice. We first transfer a fixed fraction of it to good surface area. Let \(\mathcal Z\) be a fixed-scale separated net on all of \(\Sigma\). Uniform gradient bounds give a fixed positive lower energy in a fixed small ball at every zero, including singular zeros. Packing, the definition of \(K\), and ball coverings show that the number of net points in \(B_{R+j+2}\) is at most \[C K(R)R^m(1+j/R)^{m+1}.\] By (41) and Modica’s inequality, the energy at points of \(B_R\) whose distance from \(\Sigma\) is greater than \(P_0\) is therefore at most \[C K(R)R^m\sum_{j\ge P_0} (1+j/R)^{m+1}(j+2)^n e^{-c_1j}.\] Choose \(P_0\) large and fixed so this is a small fraction of the initial energy. Energy within the enlarged rough collar is also a negligible fraction: its density is bounded and its volume in \(B_{2R}\) is \(O(K(R)R^{m-4+2\alpha+n\beta_0})\). Enlarge the collar by \(2P_0\) without changing this estimate. Every remaining point lies within \(P_0\) of a zero \(z\) in \(B_{2R}\) with \(m_H(z)=1\). Such bases have \(D\to0\) uniformly. Their fixed normal tubes are regular with bounded Jacobian, and cover the points in question by nearest projection. Their energy is at most a fixed constant times their base area by the area formula and the bounded energy density. Thus \[\mathcal H^m(\Sigma\cap B_{2R}\cap\{m_H=1\})\ge c K(R)R^m.\]

The large-reach part can now be removed with the jointly injective tube estimate from Remark 21, applied with threshold \(L/2\). More explicitly, its normal map is injective simultaneously over all bases with \(d>L/2\) and on both normal sides for \(|t|<c_2L\), where \(c_2>0\) depends only on the fixed penalty. Its Jacobian has a fixed positive lower bound. Since \(L\le R\), the images of bases in \(B_{2R}\) lie in a ball of radius at most \(3R\). Consequently \[\mathcal H^m(\Sigma\cap B_{2R}\cap\{d>L/2\}) \le C R^{m+1}/L =C K(R)R^m/\sqrt{K(R)}=o(K(R)R^m).\] Subtracting this from the preceding good-area lower bound proves the lemma. ◻

Completion of Theorem 2. If a solution attains either endpoint of \([-1,1]\), it is the corresponding constant by the strong maximum principle. The remaining constant, zero, is unstable. Indeed, for any nonzero \(\chi\in C_c^1(\mathbb R^7)\), the test \(\chi_R(x)=\chi(x/R)\) gives \[Q_0(\chi_R)=R^5\int|\nabla\chi|^2-R^7\int\chi^2<0\] for large \(R\). For a nonconstant stable solution with values in \((-1,1)\), if \(D\) vanishes anywhere, the equality case in Section 2 gives a planar heteroclinic. Otherwise \(D>0\) everywhere. If its centered density is bounded, Theorem 3 applies. If that density were unbounded, then \(K(R)\to\infty\) and Sections 1–3, followed by the contradiction just proved, would apply. Thus this last case cannot occur, and Theorem 3 gives precisely the claimed constants and planar profiles. ◻