A positive resolution of De Giorgi’s conjecture in dimension eight

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

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 (3.2). 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(\R^7) and a nonzero \zeta\in C_c^1(\R). Test with \eta(x',t)=\varphi(x')\zeta(t/L) and divide by L\int\zeta^2. The result is

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 1.2 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)}\dd b. For a unit vector e and a competitor w\in H^1(\Omega,[-1,1]),

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

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

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

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 (3.5).□

Proof. Compactness and the lower bound. Recall the phase primitive \Phi(a)=\int_{-1}^a\sqrt{2W(b)}\,\dd b. 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

Thus these measures are locally bounded. Moreover

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

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 (3.6). Both w_j and u_j converge on A to 2\mathbf1_E-1. Their boundedness implies

For \eps_j<\delta/4, partition all but a remainder of A into L_j=\lfloor\delta/\eps_j\rfloor-1 disjoint radial layers of thickness \eps_j. Thus L_j\ge\delta/(2\eps_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/\eps_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,

Expanding the gradient therefore yields

Averaging the right-hand side over the disjoint layers selects S_j with cost at most

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 \eps_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

Consequently, whenever \mu(\partial B_r(y))=0,

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 (3.13). These are all but countably many radii at each center. Together with (3.9), 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 (3.9), 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 (3.13) 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 [22, 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

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 [24, Corollary 3.3.1, p. 74]

The cone metric is \dd r^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

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 (3.16) 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

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 \cH^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 (5.5), 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

The enclosing radius is fixed, so the constant depends only on dimension. Integrating (5.5) against a nonnegative cutoff equal to one on B_3(z) and supported in B_4(z) gives

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 (5.4), and using D_i=-2(Tp)_i and D_{ij}=-2p_k\partial_kT_{ij}-2(T^2)_{ij}, gives the explicit system

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

The local mean-value estimate, (5.12), 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 (5.5): 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,

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 (5.9).

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

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 (5.10). The lower bound for D follows from the local Harnack inequality already established. Finally apply (5.10) 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 (5.11); 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,

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

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

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 5.1 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

give \mathcal T e_n=0 and \partial_s\mathcal T=0 after division by \sqrt{\epsilon_j}. Equation (5.4) gives \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

Conjugation by G, using G'/G=-\sqrt2 g and G''=(3g^2-1)G, gives (5.17). Finally (5.15), G(0)=1, and G'(0)=0 give (5.18).□

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 (5.19). This construction does not interchange an uncontrolled source integral with a limit.

The homogeneous extreme points. It remains to control h in (5.19): 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\R^{n-1}, uniform Harnack chains give

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 \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

A positive solution f gives the ground-state identity

Use compactly supported approximations to \sech y, for which (-\partial_y^2-2\sech^2y)\sech y=-\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 (5.23) 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

The second member dominates with opposite signs at the two ends, so a positive solution is a multiple of \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

Integration by parts, with boundary term zero because a<4, gives

Thus the ratio of half the derivative to this mass is

The function M is convex. Its value and derivative at 2 can be computed without special-function formulas: with t=(1+\tanh y)/2,

For the last equality the two logarithmic integrals are -1/9 and -11/18, respectively. The tangent-line inequality for convex M therefore yields

because the difference in the second inequality is (a-2)(a-4)/(6a)\ge0. The ratio in (5.24) 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

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

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 (5.21). It implies (5.20) 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 5.4 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 (5.27). Apply Lemma 5.3 to translate and rotate them. On the fixed interval [-P,P], smooth convergence gives

The source in (5.17) is nonnegative and even in s, so the limiting mean satisfies the same bound with coefficient c_1. The violated inequalities and (5.18) 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 first obtain the critical solution and uniform information at its bad centers. We then transfer that information to every distant good zero, where the later gap test will be applied.

Attainment and the bad set. Near-unit-density rigidity, stated in [10, Theorem 2.16] and originating in [25, Theorem 11.1], makes 1+\delta(n) subcritical. If some finite density is not subcritical, the finite critical density is attained by [10, Proposition 3.6]. Take the bad set in its Definition 3.7: its centers satisfy |v|\le0.9 and the reverse nonstrict inequality in the good-ball curvature condition above. Lemma 3.8 there makes this set nonempty. It is harmless to take its closure; all estimates below pass to that closure by continuity and shifted-ball containment. Propositions 3.9–3.10 there give uniform large-scale planar confinement, uniform convergence of density at these centers, integrality of the critical density, and the no-gaps conclusion. Thus K\ge2. Lemma 5.1 and Proposition 5.2 there give the asserted sparse bad-set estimate. These results explicitly assume n\le7, including Section 5 of that paper.

For clarity, the enlargement from 1 to a fixed h costs only a constant depending on h,n. Choose a maximal separated family of bad centers in B_{r+h+2}(y) with fixed small separation. Disjoint balls of a still smaller fixed radius about its centers lie in B_1(Z)\cap B_{r+h+3}(y). Balls of radius h+2 about this family cover B_h(Z)\cap B_r(y). This proves the claim using the estimate at the slightly larger radius.

Exactly K sheets at every distant good zero. We now prove (ii), with the same moduli at all good points. Choose a nearest y\in Z to z, with s=|z-y|. There are no bad centers in B_{s/2}(z), so the good-ball theorem gives the sheeting assumptions on B_{s/4}(z). The plane supplied by large-scale flatness at y, at radius 4s, passes within o(s) of z, and the zero set in this ball is confined to an o(s)-slab about it. Translate the plane to pass through z.

Here is the precise no-gaps transfer. Write A_0=4s, let P_{y,A_0} be the plane through y given by large-scale flatness, and let a_0 be the orthogonal projection of z to that plane. There is one modulus \omega, independent of the bad center y, such that

The second inequality is exactly the conclusion of [10, Proposition 3.10]; it applies because a_0\in P_{y,A_0}\cap B_{A_0}(y). For any fixed \theta>0, the number b_s=\theta s-|z-a_0| exceeds A_0\omega(A_0^{-1}) for all sufficiently large s. Monotonicity at a_0 and B_{b_s}(a_0)\subset B_{\theta s}(z) then give

All thresholds and errors are uniform in z,y; no diagonal selection of centers or radii is involved.

Set a=s/4, the radius of the ball where the sheeting assumptions hold. Translating P_{y,A_0} to pass through z confines every zero in B_a(z) to a slab of width at most 2A_0\omega(A_0^{-1})=o(a). Apply [10, Theorem 4.8] on B_a(z). It represents all zeros in the cylinder B'_{3a/5}\times[-3a/5,3a/5], in the resulting coordinates, by N ordered graphs defined over B'_{3a/5}, each with height in [-a/8,a/8]. Thus every listed graph meets B_{a/2}(z), and no zero in that ball is omitted. Its Lemma 4.9 gives, for any prescribed sufficiently small \delta_1>0 and all sufficiently large s,

The preceding density transfer with \theta=1/8 gives M_{a/2}(z)\to K. First \delta_1\le1/2 gives N\le2K; then choose \delta_1<1/(8K) and a density deficit smaller than 1/4. It follows that |N-K|<1/2, and hence N=K. The curvature, separation and profile conclusions are those of the same results. All applications use fixed interior fractions and uniform moduli.

Uniform disappearance of discrepancy. Integrate the first term of (2.3) between r and 2r. Its nonnegativity gives

Uniform convergence at bad centers proves (iii).□

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(\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 (5.9) also gives the first inequality of (6.2). To obtain its second inequality, apply (5.8) 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 (2.3) gives

In a unit tube, |p(\Phi(z,t))-\nu(z)|\le C\sqrt{D(z)}, by (5.9); 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

On a dyadic shell of radius a, (6.2) bounds the first error by Ca^{-2}, and the area bound does the same for the second. The mean-curvature term in (6.3) is at most Ca^{-1} on that shell. Summing proves the explicit uniform bound

which tends to zero by uniform density convergence at Z. The constants can be fixed when the unit tube and (6.2) 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:\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

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 (6.4), first observe the exact identity

Together with Lemma 6.1 this gives, uniformly at bad centers,

Its radial distributional consequence is

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

Exponential decay (5.14), 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

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 (6.4) 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 (5.8) 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

Stieltjes integration by parts in (6.7) gives

for fixed b,c>0. The same leading term holds after including the cutoff shells. Division by \sigma\log R proves (6.5).□

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

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

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 (6.8).

The Jacobian and the count of two. The exact projection area Jacobian is

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

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 (6.10) 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

Thus the error between \Delta_\Sigma\phi_R and its flat m-dimensional radial Laplacian is at most

It contributes at most T_0\eta(R_0), by (6.3).

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

On a dyadic shell of radius a, (6.6) 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 (6.11). 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 (6.11) is at most C(1+\log R). Indeed |H|\le CD, and each dyadic shell contributes O(1) by (6.2). For its foot term use (6.9), 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 6.1, an adjacent height difference h=f_{i+1}-f_i>0 satisfies

where

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

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 (6.11) 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 (6.12) in the second case, and its first case already has the asserted bound.□

Completion of the proof of Theorem 1.3. The nearly linear gap test. Fix a small tolerance \delta>0 and use

Because l\to\infty, \psi'\le1+\delta once S is large. Moreover

by (6.12). Thus (6.10) and Lemma 6.4 apply with L\le1+\delta.

The upper logarithmic coefficient. Let 1/(2\sqrt2)<c_*<\sqrt2/3 and P be those of Lemma 5.6. Increase S so this lemma applies on \Sigma_{S/2} and its P-tube is injective. The source term of (6.11) contributes one curvature integral; the target terms contribute at most two by Lemma 6.3. The area Jacobians and radial-weight changes cost a relative 1+C\delta after increasing S. Thus the curvature integral in (6.11), including its factor L, is at most

At every such base point,

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 (6.5) therefore gives

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,

when S is large. The endpoint normals have opposite orientations and D=o(1), so |p(z)-p(\pi(z))|\ge c>0. Apply (5.9) 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

Squaring proves (6.14).

Choose \gamma>0 so \beta=(\sqrt2+\gamma)(\sqrt2-\delta)<2. On the exceptional set

(6.14) gives D(z)+D(\pi(z))\ge cR^{-\beta}. By (6.2), the area formula, multiplicity two and the radial comparability of the feet,

Outside this exceptional set on the support of F, l\ge T_0; since l>1, w=T_0. Applying (6.4), with its error chosen at most \delta, gives

The incompatible coefficients. All cutoff errors in (6.11) 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 (6.13). Cancellation of the positive factor K m\omega_m(m-2) yields

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 5.6 is essential. No critical solution exists, proving Theorem 1.3.□

Proof. Theorem 1.3 makes every finite K_f subcritical. Apply [10, 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 [10, 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 (5.14) would bound the energy by C e^{-c\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 (5.9) 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 (7.3) 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 (7.3), the zeros in B_{3R} lie in |x_n|\le C R^2/d. With \mathbb S=e(v)\Id-\nabla v\otimes\nabla v, the stress tensor identity \operatorname{div}\mathbb S=0, tested against x_ne_n\zeta^2, gives

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

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

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 (7.5); 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,

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 (7.4) 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

since (g')^2b'=\int_t^\infty g'(r)\,\dd r. Both terms in the bracket in (7.6) 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 7.3 is subcritical by Theorem 1.3. Apply Corollary 6.6, 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 (7.4) makes this ratio as small as required, without changing the density bound. The resulting graphs have arbitrarily small fixed slope, the first estimate in (7.7), and

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 (7.8) 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

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

For the second assertion use (7.8) 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

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 \Id+tA, the graph metric, and (7.9) give, with derivatives through order 20,

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 7.4 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 (7.10) 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

The coefficient norms needed for this bootstrap are uniform by (7.11); 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(\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

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 (7.10) minus Hg', the exact equation on the whole normal line is

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

We normalize kernel coefficients by \int(g')^2. From (7.11) and (7.12), in L_t^2 with y-derivatives through order 11,

Here t^2\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

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 (7.9); derivatives of the remainder in (7.13) are already smaller. Applying the inverse estimate to the entire orthogonal component consequently gives

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 (7.15) 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 (7.10). Positive derivatives through order 6 of the even part of E now obey

Indeed the odd term -t|A|^2g' still contributes nothing; a positive derivative of \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 (7.15) 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 (7.14) 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

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 (7.18): 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 (5.9) 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 (7.17).

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,

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

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

For the last inequality integrate D\le2(1-\partial_sU) along [0,P/2+1], use (7.18), 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 (7.18).

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\,\dd S. Comparison with potentials one and two gives

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 (7.20).

Combining the two bounds with G(P/2)\le Ce^{-\sqrt2 P/2} gives

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 (7.17).□

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

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

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 7.6 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=\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

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

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 (7.3), 0\le M\le (1+r\max_i|\lambda_i|)\Id, so

Dividing the preceding second derivative expression by rs^2 therefore gives exactly (7.25). 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 7.5 and 7.6. 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

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 (7.25) 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 (7.22) 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

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 (7.21).

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

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

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

The Euclidean hypersurface identity

follows by commuting the two derivatives in Codazzi and applying the Gauss equation. This is the Simons curvature calculation [23, 24], with the mean-curvature terms retained because the sheet need not be minimal. More explicitly the Ricci contraction is ((\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

Differentiating 2/f now gives

The last term is C d^{-\gamma}(a^2d) by Lemma 7.5, h=d^{1-o(1)}, and a^2d\ge4/d. This proves (7.22). 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 (7.21) 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

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

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:

This can also be derived directly from \dd\ell=e\cdot\dd w and \dd s=\dd N\cdot e+(N-se)\cdot\dd w/\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\eps_1, the algebraic terms, including -\ell\Gamma'sa^2, are at most

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\eps_1a^2d and C\eps_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 \eps_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 (7.27) are O(d^{-1-\gamma}) from above by the base and second-foot estimates. They are absorbed exactly as in (7.25), proving (7.22), including its small-ad statement. The gradient formula, \ell\le d, and da\le2\sqrt m give a fixed bound in this range, hence (7.21).

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 (7.23). 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 (\dd\ell)N_y and leaves \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 (7.24). 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,

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 7.1 and Lemma 7.6. The proof of Proposition 7.7 supplies the corresponding tangent balls at second feet, with radius measured by the source reach d(z). Exponential comparison in (5.9) 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<\eps<1, since decreasing \eps strengthens the conclusions. The two-sided comparison with D follows from the local Harnack bound in (5.9), 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 5.6. To prove the additional curvature estimate, and to identify the common limiting quantities, consider the normalized limits of Lemma 5.3, 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

We use the nonnegative source part of Lemma 5.4 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

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

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-\eps)a(z_j)^2. The lower local mass bound gives a(z_j)^2/D(z_j)\ge c/(1-\eps)>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 5.6 with coefficient 1/(2\sqrt2)+\eps, 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 (5.9). 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 5.4 give

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,

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 \eps and apply Cauchy–Schwarz. Since F(x',0)=\mathcal D(x',0), this proves the limiting version of (8.2). Smooth normalized convergence and the uniformly positive local mass of \mathcal D^2 allow a fixed strict relaxation. The graph area elements converge to \dd x', 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

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 (8.3), 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 8.3. 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

The exponential remainder is o(D) uniformly as D\to0, since a^2\le CD.

The elementary-symmetric expansion of J and evenness give

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

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,

To prove it, initially |p(z)-\nu|\le CD(z), and along the segment

A Gronwall estimate gives (8.5). Here Tp=-\nabla D/2 and (5.9) were used.

For a fixed segment [-Q,Q], (8.5) 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

This expression is nonnegative on both half-lines. Integrating and using Tonelli, its total mass is \tfrac12\int_\R G^3\mathcal D\,\dd s. 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)\,\dd t. For Q\ge P, integration by parts gives exactly

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

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 (8.4). 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 7.4 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

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,

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 8.5. 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 7.1.

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

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

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 (8.7). 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

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

The last inequality uses D\le2(1-p\cdot n_0). The positive kernel of -\Delta+2 is

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 (8.8), 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 (8.9) gives

This proves the second inequality of (8.6), 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

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

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

Lemma 8.6 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

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 (8.10) gives exactly

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

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,

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

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^*_+,

Sign-ball clearing, as proved with (5.14), 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

is positive and at most k\eta_0. With r_j=C_je^{-a_0l_j},

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 (8.13).

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

For some fixed \gamma_1>0,

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

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 7.6, or its explicit inequality (7.18), 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 (8.7). 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 (8.14). 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

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

These bounds also hold on the fixed compact initial interval. If c'=n'\cdot e_n, then

By (8.15) 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 7.4 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, (8.13) 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 (8.14). 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 (8.12).

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 (8.15); 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 (8.11) and (8.12) show

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 (8.6) 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 5.1, applied on a fixed regular graph contained in the good set.□

Proof. Multiplying the first inequality of (8.6) 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 \dist(w,X)>2. On its fixed regular graph, for w' within intrinsic distance \mu, Lemma 8.7 gives

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

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

The surface measures here are restricted to \Sigma\setminus X. By Lemma 8.7 and \chi_R(z)\le e^{|z-w|/R}\chi_R(w), for R\ge2/\lambda,

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 (8.6) and \lambda have been fixed. This proves (8.16). 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

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

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

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 (8.17). 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

Its intercept is therefore within (C_0+1)\log N of z for large N. Applying (8.17) 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

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 (5.9) 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 (8.18) let \ell=\ell_{\min} and fix n_0=\nu(z) in (8.9). On the column z+s\nu(z), |s|\le P, (8.5) and local Harnack imply

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 8.9 and (8.5) at its bases. Thus the exact source on the right of (8.9) satisfies

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

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

Choose P,Q large and fixed, then N large. The elementary substitution r=e^{\sqrt2t} gives

The limiting coefficient is therefore

which proves (8.18).

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 (8.19), 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 8.9 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 (8.13). 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 8.9 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

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

Finally q=(1-v)(1+v)\le2(1-v); take \eta sufficiently small in terms of \eps to prove (8.19).□

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.

The Kato gain. We first record the algebra behind the strict gradient gain. If A is a symmetric tensor and \nabla A is totally symmetric, there are dimensional \theta>0,C such that

For |A|=|\nabla A|=1, equality in ordinary Kato requires \nabla_iA=b_iA. Total symmetry, after placing b in the first coordinate direction, forces A to have only its 11 component. Consequently \nabla H=b\tr A\ne0. Compactness of the normalized finite-dimensional tensor sets therefore gives a strict Kato gap when |\nabla H| is sufficiently small; when it is larger, the last term in (9.6) pays for the inequality. Homogeneity proves the claim. The weak gradient of |A| is zero almost everywhere on its zero set, so this includes that set.

Simons’ identity away from bad zeros. Choose the averaging radius S in (8.2). Let m_c be a Lipschitz cutoff vanishing within distance 3S of X, equal to one outside distance 6S, and with bounded gradient; take m_c=1 if X is empty. Put u=\chi m_c. The hypersurface identity, with our convention A=\dd\nu, is

Pair with A u^2 and integrate by parts. Codazzi gives \operatorname{div}_\Sigma A=\nabla_\Sigma H. Thus the term with \nabla^2H costs at most C\int u^2|\nabla H|^2+C\int a^2|\nabla u|^2; the cross from \Delta A uses \langle A,\nabla_iA\rangle=a\nabla_i a and costs an arbitrarily small fraction of \int u^2|\nabla a|^2 plus C\int a^2|\nabla u|^2. Since |H|\le CD, (9.6) yields, for a dimensional \theta'>0,

Applying (8.4) to au, and using the curvature lower bound in (8.1) with sufficiently small tolerance, gives

Subtracting this from (9.7) controls both G and F by the remaining errors.

Paying the mean-curvature and collar errors. It remains to control the errors in (9.7) without losing the small coefficient of D^2. For every center in this application there is one uniform sheet on B_{4S}(z). Fubini in (8.2) therefore gives

The averaging balls do not meet X. For each fixed point counted by the integral, its centers lie on one chart of area at most CS^m; the factor S^{-m} cancels this bound. The ratio of the two weights is at most e^{2S/R}, hence at most two for sufficiently large R. This explains why the constant multiplying \eps is independent of S. Furthermore a^2\le CD implies Da^3\le C\sqrt\delta D^2. The cutoff cost is bounded by

The discarded part of A_4, where m_c\ne1, is also C_SN_R by (9.4). Choose \eps' small enough to absorb, then the tolerance in (8.2) and \delta small enough for the prescribed \eps_2. Equation (9.3) proves (9.5). All integrations may first have compact support; the exponential weight, polynomial area bound, and uniform unit derivative bounds remove the additional cutoffs. Approximation of a by (a^2+\varepsilon^2)^{1/2} justifies its use as a test.□

Proof. If X is empty, apply (8.4) 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 (9.8). After estimating those terms and using the mass reserve, only the two partition terms remain to be canceled by the variance identity.

Construction: a collar on which f=1. The base tangent-ball estimate in Corollary 7.8 and (5.9) give a uniform bound

Indeed a nearby x\in X gives D(z)\ge c\delta e^{-C(10L_0+1)}, whereas a base tangent ball gives D(z)\le C_\Gamma/d(z) when d is large. The same bound, with a changed constant, applies to the source reach d(z) of a plateau pair (z,y) whenever s_X(y)\le10L_0, because (7.24) gives D(y)\le C_\Gamma/d(z). Choose t_* to cover both bounds. This choice will also be used in the multiplier argument.

Choose Lipschitz functions on physical space

On \Sigma_g choose \mu=0 for s_X\le L_0/2 and \mu=1 for s_X\ge L_0, with |\nabla\mu|\le C/L_0, and put p_0(z)=\mu(z)f(d(z)). Use the normal immersion \Phi(z,s)=z+s\nu(z) and the even curvature cutoff \beta(z,s) from Lemma 8.3, supported strictly inside |s|a<\alpha and equal to one for |s|a\le\alpha/2. The parameter \alpha is fixed small in terms of \tau before shrinking \delta. The physical ambient function and the lifted functions are

They define the single physical test F=(\sum_j\xi_j^2)^{1/2}. Here the sum includes \xi_0 and every nonzero lifted preimage of the physical point; no global numbering of the sheets is chosen.

Admissibility and the full variance. The immersion need not be globally injective. To justify the physical test without choosing global sheet labels, first cap |s| and physical support by smooth cutoffs. The nonzero lifted support is then compact inside the immersion domain: |z| is bounded because both |x| and |s| are bounded, and \mu keeps the bases away from X. Fibers are discrete by local invertibility and finite by compactness. Around any physical point finitely many local inverses therefore contain all nonzero branches. A smaller neighborhood has no additional branch arriving from elsewhere in the compact support. Thus F is locally Lipschitz, including where it vanishes, and is a compactly supported stability test. Almost everywhere,

The potential terms add exactly, even where 3v^2-1 is negative. Consequently the physical quadratic form is the sum of the ambient and lifted forms minus \int\mathcal V. No partition of a chart is introduced. The area formula accounts for every lifted preimage. Moreover the variance of any subfamily is a lower bound for \mathcal V, since each omitted square is nonnegative before minimization.

For each fixed normal cap remove the spatial cap, and then remove the normal cap. For the lifted terms this is dominated convergence in (8.3), using

For each fixed f the products |f'|\,|\nabla d| are bounded, and normal cutoff costs are polynomials in |s| times G(s)^2. Polynomial good-area growth and the exponential spatial weight dominate the remaining factors. For the subtracted variance retain only the lower bounds on fixed short segments used below, and apply Fatou’s lemma. This proves that the same stability comparison is valid after both caps are removed. The dominating bound here may depend on f; it is used only for convergence. No coefficient in the quantitative estimates below uses its value.

Claim 1: a positive mass reserve independent of f. The favorable terms will pay for the transition collar while retaining the full stated strength outside it. Define

Disjoint fixed small balls about the unit-separated bad net yield c_\delta N_R from the term \int q^3D\chi^2\alpha_0^2: at their centers q=1 and D\ge\delta, and unit regularity and (5.9) give uniform lower bounds in the balls. These balls are disjoint from a fixed small normal tube over S_{\rm sw}. That tube is injective: small defect gives a single graph in a uniform fixed neighborhood of every base, and any overlapping short normal segments must be in such a chart, where the normal map is injective. In this tube the ambient term gives

Here q^3D is uniformly comparable to D(z), and freezing \alpha_0 and \chi accounts for the two errors.

The lifted potential and Jacobian terms of (8.3) provide the complementary positive mass with b(z)^2, and on s_X\ge4L_0 provide (1-\tau')\sigma\int S_0f^2\chi^2, where \tau'>0 is as small as prescribed. To check the freezing error, on the cutoff tube

because |H|\le CD and a^2\le CD; expand the determinant in powers of sA. The nonpositive potential term can be restricted to the fixed finite segments used to extract I_P. Freezing b^2,\chi^2 on these terms therefore costs at most C L_0^{-1}D_{\rm sw}+o_R(1)D_{\rm sw} on S_{\rm sw}. Outside S_{\rm sw} any change of b between a supported base and its image requires

Equation (9.4) and exponential decay of G bound the resulting integrals by o_{L_0}(1)N_R, uniformly for large R. The curvature cutoff errors Ce^{-c/a}f^2 are absorbed by the available I f^2 on shrinking \delta. Here is an explicit mass reserve. Let c_{\rm amb}>0 denote the ambient switch coefficient, take the pure-sheet loss below \tau/16, and fix

On S_{\rm sw}\cap\{s_X<4L_0\} the complementary ambient and sheet gains supply at least 2c_{\rm sw}(\alpha_0^2+b^2)D=2c_{\rm sw}D. On s_X=4L_0, retain (1-\tau/2)\sigma S_0f^2 for exterior strength. Since b=f=1, the spare sheet gain is at least (7\tau/16)\sigma S_0\ge(7\tau\sigma c_I/32)D, which also pays 2c_{\rm sw}D. This leaves c_{\rm sw}D_{\rm sw} for errors and the desired exterior strength after the remaining tolerance losses. This treatment includes the level set s_X=4L_0, even if it has positive surface area. We have therefore reserved positive fixed multiples of both D_{\rm sw} and N_R; their coefficients were chosen before f. The next two claims show that this reserve suffices.

Claim 2: every derivative cost except the partition terms. For clarity we expand every differentiated amplitude in the lifted gradient term. Let B_s=(\Id+sA)^{-1} on tangent vectors and write \beta=\beta_0(sa) with \beta_0 even. Almost everywhere,

This includes the derivative of a; its weak gradient is uniformly bounded by the unit derivative estimates on good charts. At a=0 the derivative of this cutoff is zero almost everywhere. All vectors in the following expansion are physical vectors at x=\Phi(z,s):

Their sum is \nabla(\chi b\beta\mu f). Keep |V_b|^2+2V_b\cdot V_\chi untouched, together with the ambient partition terms. These are the following ambient and lifted densities, respectively:

For a fixed arbitrarily small \varepsilon>0, all other squares and crosses are bounded by

The vector V_b differentiates the physical partition. Its displayed square and its cross with V_\chi must keep their signs for the later cancellation; all other terms can be estimated now. Every lifted integral below has the factor G^2J. The V_f square uses the inverse metric bound 1+O(\alpha) and the normal integral \sigma+O(R^{-1}). The V_\chi square uses |\nabla\chi|\le\chi/R. The V_\mu square is a long tail: s_X(z)\le L_0 on its support, while b(x)\ne0 forces s_X(x)>2L_0, hence |s|\ge L_0.

For V_\beta, the derivative support has |s|a\ge c_\alpha. Polynomial factors in |s| therefore give a normal integral at most C_\alpha e^{-c_\alpha/a}. Freeze b(x)^2 at b(z)^2 as above: the frozen contribution is C_\alpha e^{-c_\alpha/a}f^2b(z)^2\chi(z)^2, absorbed by the sheet gain since e^{-c_\alpha/a}\le o_\delta(1)a^2\le o_\delta(1)D. The freezing error on S_{\rm sw} is at most C L_0^{-1}D_{\rm sw}; off it, the error is a long tail bounded by o_{L_0}(1)N_R.

Cross terms with the partition derivative. Only the three displayed crosses against V_b remain. If V_b\cdot V_f\ne0, then f'(d(z))\ne0 and x is in the physical switch. The collar plateau gives s_X(z)>10L_0, so |s|>7L_0. Young’s inequality pays \varepsilon|V_f|^2 from the exact derivative energy and leaves C_\varepsilon|V_b|^2 on this long-ray support. It is a tail controlled by f\le1, with no bound on f' required. The cross V_b\cdot V_\mu is a long tail for the same support reason as the V_\mu square. Finally, on S_{\rm sw} the absolute V_b\cdot V_\beta integral is at most C L_0^{-1}\int_{S_{\rm sw}}\chi^2e^{-c_\alpha/a} \le C L_0^{-1}D_{\rm sw}; off that set it is again a long tail. All other crosses, including those between V_f and V_\beta, were included in the first Young bound.

Consequently the costs apart from (9.11) are at most

plus arbitrarily small fractions of the favorable mass already retained. Every occurrence of f' has been assigned to its displayed derivative energy. All remaining bounds use 0\le f\le1, so these constants are independent even of the finite slope bound C_f^{(0)}, as well as of the later bridge parameters.

Claim 3: cancellation of the partition costs. The preceding estimates have assigned every occurrence of f' to its exact derivative energy. We must now cancel the two signed costs in (9.11), rather than spend a fixed amount of the surface mass on them. The required estimate is

The first integral includes the lifted measure for the second density. The error is uniform for R sufficiently large and for the entire class of f in the lemma. We supply the details, including the order of the logarithmic cutoffs. We will cancel the ambient partition cost against one lifted cost at each physical point, so the choice of that lifted preimage must be unique. The remaining short-ray costs will be bounded by defect mass, and long rays by the bad-net mass.

Short rays and uniqueness. Let \mathcal W=\{2L_0<s_X<3L_0\}, choose M=C_1\log L_0, and temporarily fix a small \xi>0. A preimage (z,s) of x\in\mathcal W, |s|\le M, is called accurate if

where the fixed positive b_\xi will be small. For L_0 large, M<L_0/2, so every such base lies in S_{\rm sw} and p_0=1. On an accurate segment the diffuse estimate with exponential loss \xi/2, and Tp=-\nabla D/2, imply

In detail, along the segment |T\nu|\le CD+C\sqrt D\,|p-\nu|, |p(z)-\nu(z)|\le D(z), and \int_0^{|s|}\sqrt{D(z+t\nu)}\,\dd t\le C_\xi\sqrt{b_\xi}. Gronwall followed by integration proves (9.13). Also |s|a(z)\le C\sqrt{b_\xi}|s|e^{-(\sqrt2+\xi/2)|s|}, so \beta=1 if b_\xi is sufficiently small.

There is at most one accurate preimage of a physical point. Suppose (z,s) and (\widetilde z,\widetilde s) are both accurate. Comparing each signed depth with U(x) first gives |s-\widetilde s|\le C_\xi b_\xi and comparability of their residual exponentials. Comparing their normals with p(x) and using z=x-s\nu(z) next gives

Choose b_\xi so that these bases belong to one uniform small graph. On it |A|\le C\sqrt{D(z)} by local defect comparability. Tangential projection of the equality of the two normal images yields

The left side is at least .9|\widetilde z-z|, whereas the coefficient on the right is at most C_\xi\sqrt{b_\xi}<.9. Thus the bases and depths coincide. This proof uses signed depths and oriented normals and therefore covers preimages from opposite sides.

Cancellation along the unique accurate ray. At an accurate pair, \nabla s=\nu(z) in the inverse normal chart, and (9.13) gives

Retaining only the ambient function and this lifted function in (9.10), the variance is at least

The two pure partition squares, in physical measure, equal \chi^2(q^2b^2+G^2\alpha_0^2)|\nabla\theta|^2. For A=q^2, B=G^2, their coefficient minus the leading variance coefficient is exactly

Expanding the variance and dropping its remaining negative square, then combining the two \nabla\chi crosses, bounds the uncancelled cost by

There is no multiplicity factor for this cancellation, by the accurate pair uniqueness just proved. Other preimages can only increase the full variance.

The remaining rays. The cancellation has used at most one lifted branch at each physical point. We now bound every other partition cost by defect mass or by the bad-net mass. For inaccurate lifted preimages of depth at most M,

If an ambient point has no accurate preimage, select a closest zero. When its distance is at most M, that zero is regular and belongs to S_{\rm sw}, since X is more than 2L_0 away. Its ray is normal and inaccurate, whether or not it belongs to the curvature cutoff. Indeed an accurate full normal ray has \beta=1 by the bound on |s|a(z) proved above. Its base in S_{\rm sw} has \mu=f=1, so it would supply an accurate lifted preimage, contrary to the case under consideration. The phase estimate (5.14), with exponent loss at most \xi/2, gives

This ambient integral is bounded using the area formula for the full normal map on |s|\le M. One may restrict it to any measurable closest-point selection; the resulting integral is no larger than the integral over all eligible rays. The normal Jacobian is at most C(1+M)^m. Its critical images have zero volume by the area formula, so no inverse chart or unique nearest point is required for this estimate. Measurable selection exists since the zero set is closed and nearest points form a nonempty compact set locally.

If the closest distance exceeds M, use phase decay and the volume bound in (9.4). For these long ambient tails use one fixed coarse exponent in (5.14), independently of \xi; only the short-ray comparison above needs an exponent loss chosen in terms of \xi. For lifted rays of depth greater than M, a contribution in \mathcal W requires s_X(z)\le3L_0+|s|; the same collar bound and the exponential tail of G^2 apply. Thus, for fixed dimensional N and c>0, independent of sufficiently small \xi and of C_1, all costs in this comparison are bounded by

The Jacobian, the integration over depth, and derivatives of \beta only alter the fixed power N. Weight shifts cost e^{2|s|/R} and are absorbed in the tail exponent for all sufficiently large R.

Fixing the logarithmic window. The order of choices is now decisive. The defect threshold and its positive bad-mass coefficient c_\delta have already been fixed. Choose C_1>(N+3)/c, then \xi>0 with 2\xi C_1<1/4, then b_\xi>0 as above, and finally let L_0 be large. Both coefficients in (9.15) tend to zero; the first is bounded by a constant times L_0^{-3/4}(1+\log L_0)^N and the second by a constant times L_0^{-3}(1+\log L_0)^N. They can therefore be absorbed by the already fixed positive multiples of D_{\rm sw} and N_R. This proves (9.12) with a fixed threshold \delta.

Conclusion of the gluing estimate. Claim 1 supplies the positive reserve, Claim 2 leaves only the signed partition costs, and Claim 3 absorbs those costs with the variance. Stability of F and division by \sigma therefore give (9.8). The remaining C_{L_0}R^{-2}N_R is absorbed for R large; the interface tolerances and Young constants were chosen with room inside \tau. This also proves the asserted uniformity in the center and in f.□

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

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,

The main power range and the exponential tail. On the main polynomial bridge its polynomial contribution is therefore at most F_0 A/t in P_*/f^2, where

The tail value Q_e=P_*(t_e)/f(t_e)^2 obeys

The same upper bound holds for P_*(t)/f(t)^2 for t\ge t_e. Propagating this terminal flux backwards along the polynomial range adds at most

to P_*/f^2. Its product with t tends uniformly to zero on [t_p,t_0] as t_e\to\infty with t_0 fixed, since 1+\zeta_0-2k=-2.05. For t_0\le t\le t_e, the total is at most F_0(A/t_0+Q_e). This is less than \kappa/4+.001 by first taking t_0 large and then t_e large: the fixed factor costs less than .000354 since \kappa\le\sqrt2, leaving a positive margin for the two vanishing errors. This proves both multiplier bounds on the main bridge and the tail.

Small flux on the collar plateau. On the slow bridge there is no \zeta_0 drift. Up to the factor F_0, the ratio tP_*/f^2 is bounded by

Thus it remains below .79, and its value at t_s can be made as small as desired by taking \mu small and then t_p/t_s large. On the plateau tP_*(t)\le F_0 t_sP_*(t_s), proving the arbitrarily small compact flux. More explicitly, require the terminal contribution to tP_*/f^2 on [t_p,t_0] to be at most .005. The main, slow, and plateau ranges then have the common upper bound F_0^2(F_0A+.005)<.775<.79, and

All these choices can be made before selecting t_0,t_e, since the later terminal contribution can be required to be smaller than any prescribed positive error on [t_{\min},t_p].

Completing the choices. The strict inequality 2k>3 supplies the last decay requirement: after t_p, t^3f(t)^2 decreases as t^{3-2k}. Choose t_0 large enough to put it below \eps_3, in addition to the preceding lower bounds. Choose t_e afterwards; on its exponential tail the same quantity is decreasing once t_e>3/\kappa. All the required bounds are simultaneous. Finally -f'/f\le k/t<\kappa/2 before t_e, after increasing t_{\min}. Hence f^2e^{\kappa t} is nondecreasing up to t_e and constant thereafter, proving both exponential comparisons. The primitive estimate follows on the tail by integration and on the remaining compact interval by positivity of f.□

Proof. For t>L,

Since c\tau>1 for large R, the part t\le L contributes nothing to \mathcal F_\tau. Thus

The total mass of the measure f_*'(c\tau-u)\dd u 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

Integration of the resulting exponential lower bound for W_*(\tau,t+s)/W_*(\tau,t) gives the lower ratio in (10.2). 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

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

Since b^{-\vartheta}f_*=W_*/b', their quadratic expression in (\nabla_\Sigma\tau,\nabla_\Sigma d) is exactly the last two terms of (10.3), 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 (10.3); 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 (5.9). Its contribution to \int D^2 is at least c s^{-2\alpha}. Bounded overlap and (9.1) 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 (10.5) 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

Here (L+\log R+j)/L\le C(1+j) since L\ge c_0\sqrt R. For (10.6), the elementary inequality \sqrt{a+b}\le\sqrt a+\sqrt b gives instead

The last limit uses only K(R)\to\infty.

To prove (10.7), 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,

Hölder’s inequality, (9.1), 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 (10.7) with

For example the limiting value at \epsilon=0, \kappa=\sqrt2 is greater than .197, so the fixed choices above leave strict room.□

Proof. Integrate (10.3) 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

by (10.2), (10.1), 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, (7.22) 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, (7.21)–(7.22) instead bound the source by

For d\le M_0(L+\tau)\ll\rho its shell integral against \phi is bounded by (10.6), using W_*\le d^{-\vartheta} and (9.1). 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 (7.24). 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_*,

If the receiving point lies in the rough collar, transfer first and integrate over that receiving set. In fact (7.24) 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

Its shell bound is CK(s)s^{-2+2\alpha+n\beta_0}, smaller than (10.5). 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 (9.1); this is another error of type (10.6). 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 (7.24), namely one.

Spatial cutoff transfer uses, with fixed small \epsilon_c>0,

The latter term tends to zero by (10.7), 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

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 (10.2) gives

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 (9.1). 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 (10.8).□

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

Lemma 8.9 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 (8.18)–(8.19) 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

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.

Defect mass, tangential derivatives, and cutoffs.

Over supported bases the fixed tubes |s|\le P are jointly injective for large R: any intersecting segments have bases in one buffered chart, where separation excludes different sheets and uniqueness of projection excludes two bases on one sheet. They have Jacobian 1+o_R(1). The base is nearest there, \zeta=1, and \eta(z+s\nu(z))=\psi(z). Hence

In a single-projection region the gradient of \psi(z_i(x)) is (\mathrm{Id}+sA_i)^{-1}\nabla_\Sigma\psi. Its metric and volume factors are 1+o_R(1) at every relevant logarithmic depth. Where i is nearest, the distance to \Sigma equals |s|. Fixed-depth profile convergence and then the uniform phase tail in (5.14) give

For precision, first choose a large fixed depth to make the exponential tail arbitrarily small, then increase R to use profile convergence on the fixed interval. This proves a uniform limsup without assuming convergence on an expanding interval.

In a band, differentiation of the two base amplitudes gives \mathcal U\nabla(\psi\circ z_i)+(1-\mathcal U)\nabla(\psi\circ z_j). Convexity bounds its square by the corresponding convex combination of gradient squares. Both sheet distances are at least a fixed multiple of \log\rho, so the q^2 mass of a fixed-width band tends uniformly to zero. Normal parametrization from each base, using the single traversal in Lemma 10.4, bounds the integral of these tangential terms by o_R(1)\int|\nabla_\Sigma\psi|^2. Young’s inequality with a fixed small parameter on the switch derivative changes its square by a factor 1+\epsilon_s; its reciprocal factor on this tangential part is harmless. Combining all such terms gives

On the support of \nabla\zeta the phase bound gives q^2\le C\rho^{-2M(\sqrt2-\gamma)}. Also \widetilde\eta^2\le C\rho^{-4}, because W_*\le d^{-\vartheta} is uniformly bounded on supported bases and their radii differ by O(\log\rho). Ambient shell volume is at most C\rho^7. Thus the depth cost is bounded by

for sufficiently large fixed M. Its cross term uses another fixed Young factor. The only cost still needing a fixed fraction of \mathcal L is

since |\nabla t|\le1.

If either projected m_H is not one, a nonzero projected base is in a fixed logarithmic enlargement of the 4\rho^{\beta_0} collar of Z. Parametrize by the normal rays of the nonzero bases. Each base has at most two bands, with bounded widths and polynomially bounded logarithmic Jacobian factors. The integrand is at most C\rho^{-4} times a fixed logarithmic power. The collar bound in Lemma 10.2 and the summation (10.5) make this portion negligible. This argument counts rays, not globally named sheets.

On the remaining bands m_H=1 at both feet. Assign each physical point to a foot i for which the uncut amplitude V_i is larger, breaking equality measurably. This partitions the nonnegative integral (10.14); it does not redefine \eta, so no derivative of the assignment occurs. With r=V_j/V_i\in(0,1],

The last term is supported on \rho\asymp B when B is large at fixed R, and its cutoff difference is at most C\log B/B. Normal parametrization and (10.7) therefore bound its integral by C_R(1+\log B)^kB^{-\epsilon'}\to0.

One fixed base and one adjacent side.

Fix an assigned base z_i and one normal side that contributes a band. Its partner is the first adjacent graph on that side, fixed throughout the band. Let \ell_{ij} be the shortest Euclidean distance from this fixed base to that graph, and put \ell_i=d(z_i)/G_1. By Lemma 10.4, \ell_i is the shortest adjacent gap, so u:=\ell_{ij}-\ell_i\ge0. The projected foot z_j moves along its graph as the normal depth changes; \ell_{ij} does not.

We justify the additive error for the moving foot. Put z_i=0 and its normal ray on the vertical axis, so x=(0,s_i) and s_i is the actual normal depth. Write the partner graph as x_7=f(y), with common-plane slope bounded by \theta_N=o(1) at a polynomial rate, where N=\rho(z_i). The stationary equation for projection gives |y_j|\le C\theta_N\log N and f(y_j)=f(0)+O(\theta_N^2\log N). The logarithmic lower separation then gives

after choosing the positive vertical side; the negative side is identical with absolute heights. The second equality also follows by applying the same projection equation to the base z_i itself. The chord makes an unoriented angle tending to zero with both graph normals, so its penalty at z_j is on the plateau. It is an admissible receiving-reach competitor, giving

The radial factors satisfy h_j/h_i=1+o_R(1) and |\tau_j-\tau_i|\le C\log\rho/\rho. By (10.4), monotonicity in d, and the uniform divergence of the supported reaches, there is \epsilon_R\to0 such that, throughout all assigned bands,

Here both feet have m_H=1 and thus diverging reach. To verify the inequality when d(z_j)>d(z_i), integrate the logarithmic derivative in d from (10.4) from d(z_i) up to the upper bound in (10.16), and use its \tau-derivative bound to change \tau. If d(z_j)\le d(z_i), decreasing reach only increases W_*, so the same lower bound holds. The additive distance error in (10.16) enters a multiplicative factor tending to one. This proves (10.17) uniformly for \rho\ge R.

The sharp phase input and Lemma 10.4 give

Indeed e^{-\sqrt2s_i}+e^{-\sqrt2s_j} =2e^{-\sqrt2(s_i+s_j)/2}\cosh(\sqrt2t), and its sum of distances differs additively from \ell_{ij} by o_R(1). The finite-width integral is exact:

The change from normal depth to t and the normal volume Jacobian are 1+o_R(1). Therefore, after division by the defect mass at this base,

the contribution of this side is bounded, up to the displayed finite profile factors and vanishing errors, by

We used I_P(z_i)\ge31.999e^{-\sqrt2\ell_i}; replacing 31.999 by 32 in (10.20) contributes the explicit factor 32/31.999 below. A base with m_B=0 contributes no principal term in (10.15), so division by zero is never needed.

The vanishing error in this normalized assertion is uniform even if u grows with \log\rho. For any fixed U_0, the lower bound in (10.17) converges uniformly on 0\le u\le U_0. On u>U_0 both the actual loss and its proposed bound are bounded by a fixed constant times e^{-\sqrt2u}, because 0\le1-r\le1. First make this tail small by choosing U_0, then increase R. In particular the side with u=0 has loss tending to zero: (10.17) then gives 1-r\le\epsilon_R.

The count of sides now matters. There is at most one first adjacent graph on each side, and the normal ray crosses its band at most once. If both sides contribute, at least one realizes the nearest adjacent distance \ell_i; if only one side contributes and the nearest side is different, that single contributing side is the only possible positive charge. Thus at most one side needs the positive supremum of (10.20). Restricting to the measurable part assigned to z_i can only decrease the integral. No factor of two multiplies this supremum. Figure 2 summarizes this count.

The finite tolerance budget.

Since \sqrt2/\sigma=3/4, define

Set x=e^{-1.1G_1u} and a_*=\sqrt2/(1.1G_1). Differentiation of x^{a_*}(1-x)^2 shows its only interior maximum is at x=a_* /(a_*+2), while the endpoint values are zero. Hence

The defining expression is nondecreasing in G_1, so the largest permitted value occurs at 1.001, and direct evaluation gives C(G_1)<.083272621.

Use Young factors at most 1.001 for the band derivative, depth cutoff, and cutoff difference. After those fixed choices, increase R until the normal Jacobian, the inverse derivative \dd s_i/\dd t, and the exponential of the additive distance error are each at most 1.001. After all these finite factors have been fixed, the uniform estimate following (10.20) allows a normalized error of at most .002 on each side, including the nearest side, after multiplication by those factors. This is legitimate because the unamplified uniform error can be made arbitrarily small. The principal normalized switching contribution is consequently at most

Every factor is finite: the band width was fixed at 3, and all analytic tolerances were fixed before R increased. In particular the proof does not use an infinite-width switch and silently discard its tails.

Choose the depth constant M large enough for its negligible cost before the final increase of R. Then (10.12) is at least .998\sigma\mathcal L, and the sheet-gradient costs in (10.13), including the fixed Young factor for the depth cutoff, are at most 1.005\sigma\int|\nabla_\Sigma\psi|^2. The other Young factors multiply only the switch term or a tangential band term tending to zero. The rough collar and outer cutoff errors have already been shown negligible. Substitution into (5.8) and division by \sigma give

As .998-.100>.895, this proves (10.11). All estimates were made at finite B with coefficients uniform in B; the outer limit therefore precedes the limit R\to\infty exactly as required by the radial argument.□

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

By (5.14) and Modica’s inequality, the energy at points of B_R whose distance from \Sigma is greater than P_0 is therefore at most

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

The large-reach part can now be removed with the jointly injective tube estimate from Remark 7.2, 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

Subtracting this from the preceding good-area lower bound proves the lemma.□

Completion of Theorem 1.2. 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(\R^7), the test \chi_R(x)=\chi(x/R) gives

for large R. For a nonconstant stable solution with values in (-1,1), if D vanishes anywhere, the equality case in Section 5 gives a planar heteroclinic. Otherwise D>0 everywhere. If its centered density is bounded, Theorem 1.3 applies. If that density were unbounded, then K(R)\to\infty and Sections 7–9, followed by the contradiction just proved, would apply. Thus this last case cannot occur, and Theorem 1.3 gives precisely the claimed constants and planar profiles.□