Almost-everywhere regularity of stationary integral varifolds

Proof. Put t=|x-p| and let f\ge0 be a smooth decreasing radial cutoff. Testing stationarity with (x-p)f(t/r) and differentiating under the integral gives

Approximation of the indicator of [0,1) proves ball monotonicity. At a point where the rectifiable density is an integer, its limit at zero is at least \omega_m. Such points are dense in the support. Applying the conclusion at points tending to p, with radii increasing to r, proves (2.1) at every support point.

Stationarity with \varphi\nabla u shows \int\nabla_Tu\cdot\nabla_T\varphi\,\dd\mu\le0 for every nonnegative compactly supported \varphi. The preceding radial calculation, with its field multiplied by u, has the additional term

It is nonnegative: use the nonnegative radial primitive \varphi(t)=\int_t^\infty a f(a/r)\,\dd a in the weak inequality. Thus r^{-m}\int_{\mathbf B_r(p)}u\,\dd\mu is nondecreasing. At integer-density points its limit is at least \omega_m u(p). Continuity and the same approximation of the center prove (2.2). This argument uses only the weak inequality, and therefore proves the last assertion as well.□

Proof. The nearest-point derivative is D\pi=J_z^{-1}P_h. Lift X to the ambient field \overline X(q)=J_zX(y). This is the velocity obtained by moving y with velocity X and transporting z_{\rm phys} parallel in the normal connection. Compute at a point in tangent and normal frames whose respective connections vanish there. With K_z(a)=(\nabla_a A)_{z_{\rm phys}}X, the four blocks of its derivative are

The first term of the horizontal block has trace \Div_SX. Codazzi in Euclidean space gives

For completeness, Codazzi follows by writing \mathrm{II}_{ij}=(\partial_i\partial_jF)^\perp in local coordinates and commuting the three derivatives of the immersion; at the chosen point the connection terms are precisely those in the covariant derivative. Since K_z=O(|z_{\rm phys}||X|) and J_z^{-1}-I=O(|z_{\rm phys}|),

The symmetric sum of the off-diagonal blocks, paired with a\in T_yS and v\in N_yS, is exactly

Indeed it equals \langle a,(J_z^{-1}A_vJ_z-A_v)X\rangle. Its size is O(|z_{\rm phys}||a||v||X|), including in arbitrary codimension. Contracting it with P_{vh} costs C\sqrt e\,|z_{\rm phys}||X| by (2.4). Contracting the horizontal block with I-B instead of the identity gives -B:\nabla X and an error at most C|z_{\rm phys}|e(|\nabla X|+|X|). Absorb its last summand in the preceding \sqrt e term. Integrating P:D\overline X=0 and using \int_S\Div_SX\,\dd\vol_S=0 proves (2.7).

After scaling, the coefficients of |\nabla X| and |X| are respectively

Young’s inequality bounds them by C\gamma(e+\lambda). Bounded coordinate changes preserve this estimate.□

Proof. The squared Jacobian of the orthogonal projection of T onto T_yS is \det P_{hh}. Composition with J_z^{-1} proves the formula. If the eigenvalues of B are b_i\in[0,1], then 1-\sqrt{\prod_i(1-b_i)}\le\sum_i b_i=e. Also

These facts and Young’s inequality prove the bound in (2.9). Expanding the square root one order further gives \sqrt{\det(I-B)}=1-e/2+O(e^2), which proves (2.11). The rectifiable area formula [19, Chapter 3, equation (2.5)] gives the first equality in (2.10), where N counts points of nonzero projection Jacobian with their multiplicities. Its second assertion follows by pushing forward (1-J_\pi)\mu. The part on which the projection has deficient rank has e\ge1 and is included in this error; no absolute-continuity assertion about its pushforward is needed.□

Proof. First mollify on a slightly smaller domain. Choose a cutoff equal to one on B_3 and supported in B_{7/2}, and let \Phi be the fundamental solution of the Laplacian. The potential

has the same Laplacian as v on B_3, so h=v-u is harmonic there. The second-derivative kernels have the weak-type estimate |\{|u|>t\}\cap B_3|\le C\epsilon/t; the first-derivative kernels are integrable on bounded sets. We recall the kernel argument. Decompose a mollified finite measure at level t into its bounded part and zero-mean pieces on maximal dyadic cubes. The bounded part has squared L^2 norm at most Ct times the original mass norm. The Fourier multiplier \xi_i\xi_j/|\xi|^2 has L^2 norm at most one. The bad cubes have total volume at most C\epsilon/t; off their fixed enlargements, cancellation and the bound |\nabla\partial_i\partial_j\Phi(x)|\le C|x|^{-m-1} bound the L^1 norm of their contributions by C\epsilon. Chebyshev’s inequality proves the stated weak estimate, uniformly in mollification. Pairing the same potentials against a smooth test, moving the derivatives to the test, also gives |\langle u,\varphi\rangle|\le C\epsilon\|\varphi\|_{C^k} on any fixed bounded region.

The harmonic remainder is controlled modulo a constant by the full gradient equation. For x\in B_{5/2} choose a fixed radial unit-integral bump \eta_x supported in a small ball about x. Harmonic mean value and (2.12) give

Consequently \sup_{B_{5/2}}|\nabla h|\le C\epsilon. Taking c=h(0) proves both estimates for the mollified quantities. The constants c are bounded by pairing against one fixed unit-integral bump and the finite mass of v. Pass to a subsequence as the mollification tends to zero; the densities converge almost everywhere at Lebesgue points. This proves (2.13), and weak convergence against tests proves (2.14).□

Proof. Scale to s=1. Write \dd\vol_S=a(y)\,\dd y and put the factor a^{-1} in each coordinate component of the tangent test field. Then Lemma 2.2 gives an equation of the form (2.12) for the scalar measure a^{-1}\nu. The divergence data have mass at most C(D+\Lambda), as do the zeroth-order data; derivatives of the fixed coordinate factors have bounded size. In particular, no derivative of the unknown scalar mass is estimated separately. Apply Lemma 2.4. By Lemma 2.3, the difference between a^{-1}\nu and N\,\dd y has mass at most C\epsilon. The density version of this difference and (2.13) give

If \epsilon=0, this estimate for every t>0 gives N=c almost everywhere. Since N is integer-valued, c is an integer, and the distributional bound is exact; the conclusions follow directly. Assume henceforth that \epsilon>0. Taking t=C_0\epsilon with C_0 large proves that, on a fixed large majority, N is within C_0\epsilon of c. For small \epsilon there is at most one integer in this interval; call it q. Thus |c-q|\le C\epsilon. The distributional bound against the specified smooth average makes its value q+O(\epsilon), so the smooth-average hypothesis forces q=Q. The majority hypothesis has the same consequence by intersecting the two majority sets.

Take now t to be a fixed fraction of one. Since |c-Q|\le C\epsilon,

The nonnegativity of N gives \int_{B_2}(N-Q)_-\,\dd y\le Q|\{N\ne Q\}\cap B_2|. The small measure error between \nu and N\vol_S therefore bounds M_-(B_2) by C\epsilon. Choose 0\le\varphi\le1, equal to one on B_1 and supported in B_2. Equation (2.14), applied also to the bounded volume factor, gives |\int\varphi\,\dd M|\le C\epsilon. Hence

This proves total variation and the exceptional-volume assertion, and rescaling proves (2.15). Finally, on an overlap of fixed relative volume, the same fixed-threshold argument gives |\{N\ne q_i\}|\le C\epsilon_i for each ball’s selected integer q_i, before any identification with Q. These exceptional sets cannot fill the overlap. At a remaining point both selected integers equal N. Induction along a finite chain proves propagation.□

Proof. It is enough to prove the estimates for d>0. If the actual energy is zero, fix 0<\delta'<\min\{\delta_0,1/2\} and positive upper bounds d_j\downarrow0 below d_0(\delta'). Let G_j be the measurable good sets from the positive case. Fatou’s lemma applied to their complements gives

Choose a defined fiber \mathcal F_0 in this limsup and retain the indices j_k for which it is good. Its Q entries match the corresponding configurations within C_{\delta'}s\sqrt{d_{j_k}}\to0, so those configurations converge to \mathcal F_0 up to permutations. The limsup of the selected G_{j_k} has the same volume bound. On each fiber in that limsup, the matching triangle inequality along its good subsubsequence gives exact matching to \mathcal F_0. For one admissible \beta_0>0, the support estimate and the matching convergence put every support height in \mathcal F_0. All support distances and hence all finite moments are zero, proving the assertion for d=0. We give the finite selection argument before the iteration that controls exceptional fibers. If f is a bounded Lipschitz function of height with |f|\le1, use f(\zeta)\overline X in the horizontal first variation. In flat coordinates the additional term is \nabla f\cdot P_{vh}X. In normal coordinates its additional frame term is bounded by C\Lip(f)l|z_{\rm phys}||X|. In the curved case, as in Lemma 2.5, write \dd\vol_S=a\,\dd y and take the coordinate components X=a^{-1}Y. Then \Div_SX=a^{-1}\Div Y exactly; derivatives of a^{-1} remain in the tensor and geometric terms. Cauchy–Schwarz, the normalized mass bound, and Lemmas 2.4 and 2.3 show that the fiber function

differs from its fixed smooth average \overline N_f by a weak-L^1 quantity whose normalized size is at most

The same bound controls distributional pairings. Changing between Jacobian-weighted and ordinary mass in this statement costs Cd.

For this finite selection prescribe 0<\delta\le1/4; a larger requested exceptional proportion is covered by using \delta=1/4. Normalize the smoothly averaged vertical counting measure to a probability measure \mathcal P; its total before normalization is Q+O(d) by Lemma 2.5. Fix a mass mesh \eta_0 much smaller than 1/Q. For each of the n coordinates choose its quantiles at levels j\eta_0, and test smoothed half-lines immediately to both sides of these quantiles, with transition width w=K s\sqrt d. Repeated quantiles cause no problem: retain both one-sided tests. There are at most C(n)/\eta_0 tests. For each, the right side of (2.20) is C(d+K^{-1}). First choose \eta_0, then choose K large enough for these tests and the joint-count tests below, and finally choose a positive energy threshold small. The union of the exceptional sets where any test differs from its average by more than \eta_0 then has relative volume at most \delta/3.

Also discard the set where N\ne Q. By (2.16) its relative volume is O(d), so it is at most \delta/3 after decreasing the same threshold. Every remaining fiber therefore has exactly Q projection entries, counted with multiplicity.

Let F_y be the empirical coordinate distribution on a remaining fiber (normalized by Q), and let F be that of \mathcal P. Bracketing an arbitrary real number by adjacent quantiles gives

If t is in a quantile jump, the left and right tests at that quantile give the same inequalities; thus atoms require no continuity assumption. Comparison of two good fibers gives this estimate with 4w and 6\eta_0. For 6\eta_0<1/(2Q), comparison at the levels (j-1/2)/Q shows that their ordered coordinate lists differ by at most Cw. Choose one such fiber as a temporary reference. In each coordinate group its at most Q levels by joining consecutive levels whose gap is at most a sufficiently large fixed multiple of w. Each resulting interval has length at most CQw, and distinct intervals have an empty gap in which a cutoff can change from zero to one.

Coordinate lists alone do not determine vector configurations. To determine the joint counts, test products of the interval cutoffs just constructed. There are at most Q^n such tests and their Lipschitz constants are at most C/w. On the good coordinate fibers every product equals an exact indicator. After decreasing the threshold if necessary and discarding a further relative volume at most \delta/3, each joint count is within less than 1/4 of its smooth average. It is an integer, so it is the same on all remaining fibers. Taking this many copies of one point in each occupied coordinate box gives \mathcal A. The boxes have diameter Cw, proving the majority assertion. Intersecting the good sets of two overlapping balls proves the matching assertion. Write d_{\rm sel}(\delta)>0 for a threshold sufficient for this finite selection, and use d_{\rm sel}(1/4) when \delta>1/4.

There is also an exact finite selection when the energy on this ball is zero. Fix 0<\delta'<\min\{\delta,1/2\}, apply only the finite selection just proved with positive upper bounds d_j\downarrow0 below d_{\rm sel}(\delta'), and use the limsup construction and matching through one common fiber from the start of this proof. That part uses only the good sets and their matching errors, not the moment or support estimates. It yields an exact majority configuration with exceptional proportion at most \delta'<\delta.

The dyadic covering geometry below, including its enlargement and overlap ratios, is fixed from the data at the outset and determines c in the statement. Choose an internal proportion 0<\delta_{\rm int}<1/4 so that two exceptional sets cannot fill any comparison overlap, and small enough for the support argument. Then choose a fixed threshold d_{\rm int}\le d_{\rm sel}(\delta_{\rm int}) small enough for that argument and the iteration, and set

For every selection ball in the descent, including the root, use the actual normalized mass of \tau on the associated fixed enlargement as the selection upper bound, using the exact finite selection when that mass is zero. Use the root configuration \mathcal A selected with \delta_{\rm int} throughout the descent. If \delta_0<\delta_{\rm int}, apply the finite selection once more at the root with exceptional proportion \delta_0 and the same actual energy. A common good fiber matches the two root configurations within C_{\delta_0}s\sqrt d, after enlarging that constant. On the good fibers of the second selection, the triangle inequality then gives the prescribed majority bound for \mathcal A. If \delta_0\ge\delta_{\rm int}, the original good set suffices. Thus the descent and its root configuration use only the fixed internal choices; the finer selection affects only the majority constant and input threshold.

We next record how to include every support point. On a continuing ball of radius r with normalized \tau at most d_{\rm int}, the majority assertion and Lemmas 2.5 and 2.3 bound the mass outside the configuration’s height neighborhoods by

Indeed the mass above bad fibers is at most their Q-fold base volume plus |M|, and the projection error also has mass at most C(D+\Lambda). Because the restricted support is separated from the artificial vertical boundary, its extension by zero across that boundary remains stationary for fields supported away from the lateral boundary; multiply such a field by a vertical cutoff equal to one near the restricted support to verify this assertion. The monotonicity ball is taken in this extension and remains in the lateral working region; it need not lie inside the original vertical cutoff. If a support point over an inner base ball were at distance at least c_1r from these height neighborhoods, its ambient ball of radius c_2r would lie outside them. For fixed small c_2>0, this contradicts (2.1) when \delta_{\rm int},d_{\rm int} are sufficiently small. The same reasoning works in the curved tube since its coordinate map and inverse are uniformly Lipschitz. This proves an O(r) bound for every support point on an interior ball. In particular it applies to a first stopped child, using its continuing parent.

For the quantitative iteration, subdivide the inner base region into dyadic cubes, using around each cube a ball of a fixed large multiple of its side. Start with sufficiently generous interior room. Continue down a branch while the normalized \tau on these balls is at most the fixed threshold d_{\rm int}. Otherwise stop at the first such cube. On continuing balls the normalized mass is bounded by Q+Cd_{\rm int}, the integer is Q, and the majority configurations at a child and its parent match with increment

The fixed parent enlargement here contains both selection enlargements. If its \tau-mass is zero, both actual selection energies vanish, their exact majority configurations match on the overlap, and the child cannot stop. At a first stopped cube the all-support argument gives instead Cr for the distance from each support height to the parent’s configuration. Its stopping condition charges r^m to a fixed multiple of \tau on that enlargement. Configurations are regarded with their optimal matching, so their nearest-point distances obey the triangle inequality even at coincident entries.

Write r_j=2^{-j}s, and let b_j(q) be the corresponding increment for a point above a continuing cube, or its final error at its first stopped cube; put b_j=0 after a stop. Fixed enlargements of cubes have bounded overlap at each generation. The preceding charge, the bounded mass of a continuing parent, and \tau(B_{cs})\le d s^m give

For the second bound on stopped cubes, stopping itself implies 2^{jm}d\ge c>0, so its Cr_j error has the required form. For a branch which never stops, the all-support Cr_j bound tends to zero; thus its configurations approach the actual support height in distance. It follows in both cases that \dist(\zeta(q),\mathcal A)\le\sum_jb_j(q).

The covering, stopping decisions, and configuration were fixed before either exponent was chosen. The following summations therefore apply to this same \mathcal A for every exponent in the stated ranges. Interpolate (2.21) with the second part of (2.22), without the minimum, to obtain

Minkowski’s inequality and (2.17) prove (2.18). Summing (2.22) above and below 2^{-j}\simeq d^{1/m} gives Cs\sqrt d when m=1, Cs\sqrt d(1+|\log d|) when m=2, and Cs d^{1/m} when m>2. Each is bounded by C_\beta s d^\beta for the stated range of \beta, proving (2.19).□

Proof. The logarithmic derivative of G_s is -(y-x)/s^2. Its values on each fixed enlargement of a ball of the stated radius differ by bounded factors. Such a cover can be obtained by a maximal disjoint family of sufficiently small balls for this radius function; nearby radii are comparable. Apply (2.15) on each ball and sum. This proves (2.23) without a factor depending on the number of balls. Apply Lemma 2.7 with the fixed proportion \delta_{\rm cmp} successively to concentric dyadic dilations and to overlapping comparable balls. Every increment is bounded by a fixed power of t times s\sqrt d, and each local moment estimate has the same property by (2.24). Summation of the geometric increments gives the claimed polynomial bounds. Finally \sum_{j\ge0}2^{jC}\exp(-c4^j)<\infty absorbs any fixed polynomial loss in the Gaussian integral.□

Proof. In flat coordinates \dot G_s=-\Div_y((y-x)G_s). The horizontal first variation says \int\Div X\,\dd M=\int B:DX\,\dd\mu. Take X=(y-x)G_s. Since DX=G_s(I-u\otimes u), it follows that \dot M_s=-D_s+R_s. Differentiating the factor s^2 proves (2.26).

For the curved identity apply Lemma 2.2 to the same coordinate field. Its scaled derivative is bounded by CG_s(1+|u|^2) and its size by CsG_s|u|. On a physical chart of radius l, coordinate coefficients and their scaled first derivatives differ from the centered Euclidean ones by O(l). The divergence comparison is made after subtracting Q\vol_S; its coefficient error therefore multiplies M, not the background mass. The tensor coefficient errors multiply B and are bounded by C\gamma e. Equation (2.8) controls the remaining geometric terms. Multiplying by s^2 proves (2.27). Differentiation of \chi produces precisely the tails described in (2.25).□

Proof. In a product Euclidean chart, differentiate the ambient fields f(\zeta)X(y) and \varphi(y)v(\zeta). Their horizontal and vertical blocks give (2.28) and (2.29). Apply the first with f=\Gamma and X=(y-x)G_s. It yields the first term of (2.30) and the flux s\int G_sD\Gamma\cdot P_{vh}u\,\dd\mu. Apply the normal formula with v=D\Gamma and \varphi=G_s; since \nabla G_s=-G_su/s, this flux is exactly s^2\int G_sD^2\Gamma:P_{vv}\,\dd\mu. This proves the identity and fixes both its sign and its factors of s.

We give the geometric error calculation. In radial gauge the normal connection one-form vanishes at the chart center and has physical norm O(l) throughout the chart, by its bounded curvature and the radial parallel-transport equation. Its ambient derivative still has the tangential component -A_v; that component is not part of the small connection one-form. A horizontal derivative in scaled ambient coordinates of \zeta=z_{\rm phys}/l is

while the vertical derivative is unchanged. Differentiating v(\zeta) therefore gives the first summand of (2.31) on contraction with the mixed block. Differentiating its normal frame gives a horizontal trace whose identity part is -l\langle H_S,v\rangle, with error Cl|z_{\rm phys}||v| from J_z^{-1}-I; its B part is bounded by Cle|v|. The remaining mixed-frame term is bounded by Cl\sqrt e|v|. This proves (2.31). A base derivative uses D\pi=J_z^{-1}P_h; its coefficient on a coordinate gradient in the chosen frame is J_z^{-1}\mathsf C_y^{-\top}, which differs from I by at most C\gamma. This gives the stated coefficient error on the mixed block.

For the horizontal test, first use the calculation in Lemma 2.2, and then differentiate f(\zeta). If X_{\rm coord} is the coordinate column of the scaled field, its lift has column J_z\mathsf C_yX_{\rm coord} in the horizontal frame. Subtracting the flat mixed flux leaves

The bound |P_{vh}|\le C\sqrt e gives its first contribution to (2.32). The separate connection derivative is bounded by Cl|z_{\rm phys}||Df||X_{\rm coord}|, which gives the second contribution because l\le\gamma and |z_{\rm phys}|\le\sqrt\lambda. For the coordinate form, in the original centered coordinates put a_l(\xi)=\sqrt{\det g(l\xi)}. For a scaled coordinate field X(\xi),

This weighted identity has no subtracted reference volume, so this last term gives the stated Cl\int|f||X|\,\dd\mu error. The remaining coordinate coefficient changes are contracted with B and cost Cl e|f|(|\nabla X|+|X|), already included in the geometric error. At a further radius s, put y'=(y-x)/s, \zeta'=\zeta/s, T_{x,s}(y,\zeta)=(y',\zeta'), \mu'=s^{-m}(T_{x,s})_\#\mu, and v'(\zeta')=v(s\zeta')/s. Then D'v'=Dv, while \lambda remains in physical units. Keeping the same frame retains the coefficient l in l|Dv||z_{\rm phys}|\sqrt e; the terms involving v become (ls)|v'|(|H_S|+|z_{\rm phys}|+e+\sqrt e). The resulting estimates use the original \gamma. All configuration comparisons likewise use the original height frame, so there is no change of height coordinates from a new gauge. Young’s inequality changes products of two first-order errors into the stated quadratic bound. Applying Cauchy–Schwarz to (2.31) retains the actual L^2 size of v, which is why this form of the bound is useful for displacement and clipped-height tests. Finally multiply each field by the chart cutoff; its derivative terms obey (2.25).□

Proof. For \sigma\in\mathcal P(K) put g_\sigma=f-M_f(\pi_X)_\#\sigma. This measure has zero mass. Define

using the inverse with zero spatial mean. If \lambda_k are the nonzero eigenvalues of -\Delta_X, its kernel has Fourier coefficients e^{-\lambda_k s^2/2}/\lambda_k. Since s\ge a>0, the kernel and all its derivatives converge uniformly on bounded scale intervals. Thus b_\sigma is smooth, and

Weak convergence of probabilities implies uniform convergence of their potentials on K: the kernel is jointly continuous on the relevant compact product, and can be approximated uniformly by finite sums of products of continuous functions of its two variables.

Choose finite 1/j-nets K_j=\{z_{j1},\ldots,z_{jN_j}\} in K. For a probability vector p on K_j and \beta>0, the map

is continuous from the probability simplex to itself. Brouwer’s Fixed Point Theorem gives a fixed point [16, Theorem 2]. First let \beta\to\infty on a fixed net and take a convergent subsequence. If a point’s limiting payoff is at least \delta below the maximum, its fixed-point weight is at most e^{-\beta\delta/2} for all sufficiently large \beta. The limiting probability is therefore carried by maximizers of its own potential on K_j. Next pass to a weak limit as j\to\infty. Uniform convergence of the potentials and density of the nets show that the resulting probability \sigma satisfies

Indeed a compact subset separated from the limiting maximum by \delta receives no mass from sufficiently late net probabilities; a nearby net point witnesses the strictly larger payoff. Exhaust the complement of the closed maximizer set by such compact subsets.

Write b=b_\sigma. Its maximum on the whole continuation set is bounded by \max_K b. To verify this on an irregular boundary, truncate at any scale S>s_{\rm top} and apply the elementary maximum argument to b-\epsilon t. At a maximum outside K below the artificial top, \partial_t(b-\epsilon t)-\Delta(b-\epsilon t)=-\epsilon contradicts the derivative conditions. At a maximum at the artificial top, the one-sided time derivative is nonnegative and the spatial Laplacian is nonpositive, giving the same contradiction. Let \epsilon\downarrow0. This argument uses only that points outside K have an open neighborhood in the continuation set, and hence also covers arbitrary vertical-side geometry.

For (x,s_0)\in\spt\sigma, (4.4) and upward closure now give b(x,s_0)\ge b(x,s) for every s\ge s_0. The right derivative at s_0 is nonpositive. Applying (4.3) proves the first inequality in (4.2); integrating that identity proves the second.□

Proof. The only commutator is

Split at torus distance r_*=c_0\eta R. For sufficiently small fixed c_0, the near contribution is at most \eta\tau_s(x)/4. The far contribution is at most

Writing x_*=s/R, its ratio to d is bounded by Ce^{C_1T}x_*^{-m}e^{-c_2\eta^2/x_*^2}. On 0<x_*\le c\eta^2/T^2 this is o(\eta) uniformly: the negative exponent is at least a fixed multiple of T^4/\eta^2, which dominates C_1T, m|\log x_*|, and |\log\eta|. The same estimate applies to the nonnearest periodized copies. This proves (4.6).

At a contact f_{s_0}\ge h_{s_0}\ge0. If w(x)=0, (4.6) would give a strictly negative value. Division by w(x)>0, followed by w(x)\le1, proves (4.7). Integrate (4.6), use the cumulative contact inequality and the nonnegativity of the corresponding integral of h, and again divide by the fixed number w(x). This proves (4.8).□

Proof. The proof is a finite recursion over working balls, with the root configuration and its far-tilt region kept fixed. On a sufficiently small ball, the support lies in disjoint neighborhoods of the root groups. Cutting through the gaps gives stationary pieces with positive counts below Q, so the lower-integer theorem makes the signed integral small on that ball.

On a larger ball, if the signed integral exceeds a small margin, the remaining positive source gives a contact measure. The cumulative contact inequality and the Gaussian mass identity exclude a contact whose derivative scale is close to a threshold hit or a side equality: it would force too much signed mass at the outer scale. Bottom contacts are excluded by the small-ball reduction. Each remaining contact lies on a vertical side, far below its equality scale. Its signed Gaussian integral is partitioned over smaller working balls, while a choice of thresholds made before constructing the contact measure confines the associated charging balls to a small part of the measure on the working ball. Thus a large generation pays a small margin and at most half of the next-generation bound. The choices below enforce these alternatives and keep the configuration motion smaller than the original separation.

For example, after extending F_{Q-1} monotonically over a fixed initial interval, one can use

The definition of the iterated exponentials gives F_{Q-1}^{-1}((\log T)^3)=T^{o(1)}, and this choice has (4.11). All changes of b below are by fixed constants or an additional fixed power of \log T.

Choose a fixed integer N>P+10, increasing it later if necessary, and then choose a fixed J_0 sufficiently large in terms of m,n,Q,N. In particular require

and enlarge J_0 to exceed all fixed polynomial density losses in the cell constructions below. Set

The parameter T, the original configuration \mathcal A, and the measure D_{\rm far} remain fixed throughout the finite recursion. They are never recomputed at a child node.

Two elementary parameter bounds will be used repeatedly. With \ell_T=\log T, direct substitution gives

Also, for i\le n_0,

For v\ge cr_b/T^2, the logarithm of the factor multiplying \sqrt d is O_{J_0,N}((\log T)^2)+O((\log T)^{60}). Consequently O(T) such increments, even with any fixed polynomial loss in T, sum to less than e^b\sqrt d. This is why the much larger separation e^{C_Sb}\sqrt d survives every generation.

At generation i a node consists of a ball B_{4R}(x_*) with

and a majority configuration \mathcal A_R on B_{r_{\rm maj}R}(x_*), with local upper bound C r_{\rm maj}^{-m}E_{2i}(R), satisfying

The integer is Q on the balls constructing the node; the small-density and overlap checks below establish this before their configurations are selected. We estimate all weights 0\le w\le1, \Lip w\le R^{-1}, supported in B_R(x_*). Write

We prove, by downward induction on i, that

Here C_*,C are fixed, independent of i and T. The last-generation bound follows at once from |L'|\le C\tau. Initialization at R=1 follows from \tau(B_4)\le d and the given configuration.

Small nodes and actual stationary pieces. Suppose first that R\le T^{10N}r_b, and put E=E_{2i}(R)<1, as follows from (4.15). Cover B_{2R}(x_*) by a fixed finite number of balls of radius \sigma R, including the concentric ball, where the fixed \sigma>0 is small enough that every fixed enlargement needed for scalar selection and then for the support estimate in Lemma 2.7 lies in B_{4R}(x_*), and the scalar enlargement of the concentric ball lies in B_{2R}(x_*). Their normalized \tau is at most CE. The normalized smooth mass averages therefore differ from Q by O(E)<1/4, so Lemma 2.5 selects Q on these balls. On each choose the joint configuration in Lemma 2.7 with exceptional proportion \delta_{\rm maj}, retaining its majority property and using its all-support bound with a fixed 0<\beta<\min\{1/2,1/m\}. Its local upper bound is CE=o(1). Connect it to \mathcal A_R by a fixed chain satisfying (4.9). The intermediate balls have the same CE=o(1) upper bound, and the same smooth mass test selects Q there. Equation (4.10) bounds this change by CR\sqrt E, while the support bound is CRE^\beta. Consequently all support heights over B_{2R}(x_*) satisfy

for a fixed C'. The second line uses \beta<1/2 and E<1. To see the last bound without a hidden scale restriction, write R=\sqrt d\,e^{3b}T^\theta, where -3\le\theta\le10N, and substitute in RE_{2i}(R)^\beta; its logarithm after division by \sqrt d is bounded above by C'b using (4.15). Choose C_S>C'+10. For a root group \mathcal A_g, put

For large T, (4.19) puts all the support over B_{2R} inside the smaller regions with radius S(T)\sqrt d. The regions \mathcal O_g are disjoint, since the root group gap is 10S(T)\sqrt d. Thus there is an open height gap around each group’s support, and no far energy in this interior region.

The use of disjoint height regions to obtain stationary restrictions has a precedent in [5, Section 3.1]. For each group, restrict the varifold over B_{2R}(x_*) to \{z\in\mathcal O_g\}. Choose a smooth height cutoff equal to one near the smaller region of radius S(T)\sqrt d around \mathcal A_g and zero outside \mathcal O_g. Its derivative vanishes near the entire support over B_{2R}(x_*). Multiplying any ambient test field with compact base support there by this cutoff proves stationarity of the restriction. It keeps every included spatial point with its entire integer multiplicity, including on exceptional fibers. On a majority fiber of one of the local configurations, its matching to the fixed root list moves each entry by at most (i+1)e^b\sqrt d+CR\sqrt E=o(S(T)\sqrt d). An entry matched to \mathcal A_g therefore lies in \mathcal O_g. If two copies of one spatial point were matched to different root groups, their two root entries would be separated by at most twice this error, less than 10S(T)\sqrt d for large T, contrary to the separation. Each group’s projection count on that fiber is thus the number q_g of entries in its root group, including repetitions. Since there are at least two nonempty groups, 1\le q_g<Q and \sum_gq_g=Q. To specify the individual energy budgets, take a finite nonnegative partition of the weight on balls B_{r_\alpha}(y_\alpha), with r_\alpha=cR for a sufficiently small fixed c, and B_{8r_\alpha}(y_\alpha)\subset B_{2R}(x_*). These enlarged balls have bounded overlap. For each stationary restriction, its normalized projected mass is bounded by the normalized total mass, and its normalized D+\Lambda is O(E)=o(1). Apply Lemma 2.5 with its initial intermediate ball equal to the concentric majority ball above. Its selected integer is q_g by intersection with the group majority set there. Connect this ball to the intermediate balls for the scalar applications on B_{4r_\alpha}(y_\alpha) by a fixed finite chain of comparable interior balls with the positive overlap in (4.9). The scalar count exceptions on each intermediate ball have relative volume O(E), smaller than the fixed overlap for large T; on a remaining fiber the two selected integers coincide. Thus q_g is identified and propagated before the mass defect of the restriction is estimated.

Write \mu_g for the mass measure of the restriction and D_g,\Lambda_g for its projected energy measures. Now set M_g=\pi_\#\mu_g-q_g\vol_S and \tau_g=|M_g|+D_g+\Lambda_g. The scalar estimate with interior room now gives

Define the parameter for the lower-integer application of Theorem 3.1 by

Increasing the fixed C_s if necessary supplies the required bound for \tau_g on each rescaled B_4. Moreover

Here the group energies add exactly on their separated region, and the enlarged base balls have bounded overlap. Individual mass defects have been estimated by Lemma 2.5, rather than by splitting |M|. The parameters satisfy

Indeed d\le d_{g\alpha}\le d+C E_{2i}(R), and (4.15) absorbs the fixed constants. Physical radii only decrease, and physical heights are unchanged, so the curvature hypothesis follows from the original one. Apply Theorem 3.1 at q_g to the partition weights normalized by a fixed constant. Taking C_Q at least as large as the finitely many constants for q<Q, summing the lower-integer conclusions with (4.21) gives an o(\eta)\mathcal N_R bound. Indeed F_q(T_{g\alpha})\ge F_{Q-1}(b/2)\ge(\log T)^3 for large T. This argument also applies to the small cells used at a bottom contact.

Stopping tests at a large node. Now assume R>T^{10N}r_b. Use a torus of period C R, identifying the node and a fixed surrounding chart with its central portion. Put

Its total mass is comparable to \mathcal N_R. If f=wL'-\eta\nu has nonpositive integral there is nothing to prove. Otherwise apply Lemma 4.1 with bottom r_b and dyadic test scales t\in[r_b,\rho], using

The thresholds will be chosen later. There are O(T) test scales. The torus masses satisfy the growth condition in Lemma 4.2, because \mathcal N_R/(dR^m)\le\exp(O(T)) by (4.17). Also \rho/R=T^{-4N}\ll\eta^2/T^2.

For every contact (x,s_0), localization places x in the interior support region. Gaussian comparison between successive dyadic scales, and the non-strict tests at larger scales, give

Here and below the measure in this display is the node restriction. At the current mesh scale its value is bounded by the next larger scale. At the top use the node’s total mass. In fact, for v\ge\rho/16,

Thus all top tests hold strictly, and every scale with q_v\ge1 lies below \rho/16, leaving a fixed outward interval below \rho.

If s_0>r_b, the boundary description (4.1) supplies a mesh scale v_1\ge s_0 with q_{v_1}(x)\ge1. Either it is a current hit, or it is a larger scale with q_{v_1}(x)=\lambda_{v_1}. In the latter case every still larger test satisfies its continuation inequality. This follows directly from the finite intersection: away from the bottom, if there were no current obstruction and every relevant larger test were strict, a space–scale neighborhood would belong to \mathcal D.

Exclusion of a nearby hit or side equality. We show that no such v_1 can satisfy v_1/s_0\le(\log T)^{20}. All the following estimates are on this one outward vertical. Put

By Lemma 4.2, Z\ge0 and A(s_0)\ge0; the latter uses c(T)<1 and L'=2M-D-C_Q\Lambda+c(T)D_{\rm far}. The envelope gives

In particular the adverse c(T)D_{\rm far} term is small enough for the logarithmic losses below.

At v_1, and on its fixed outward dilations, the scalar estimate is available in weighted form:

Here are the raw density and integer checks, also needed for the children below. For s\in[s_0,\rho] and U=1+|y-x|/s\le C\sqrt T, put r=cs/U with a small fixed c. All required fixed enlargements B_{Lr}(y) lie in B_{2R}(x_*): x\in B_R(x_*) and C\sqrt T\,\rho=o(R). Choose a=C_1Us so that G_a(x-z)\ge c_1a^{-m} on these enlarged balls. If a\le\rho, positivity and the envelope give

If a>\rho, use the node total instead. Dividing by E_{2i+1}(s) gives the bound

In the middle inequality a negative exponent of R/s is discarded; for a positive exponent use R/s\le C U T^{4N}. Consequently, in both cases,

Since \tau includes |M|=|\pi_\#\mu-Q\vol_S|, choose a fixed nonnegative smooth mass test on this enlarged ball, normalized to reference integral one and with bounded scaled derivatives. Its average differs from Q by at most the right side times a fixed constant, hence by less than 1/4. This identifies Q directly in Lemma 2.5, and also provides its mass and energy hypotheses. Apply Lemma 2.5 at s=v_1 on a cover by these cells and sum with the comparable Gaussian weights, as in Corollary 2.8. The cutoff at C\sqrt T\,v_1 has arbitrarily small exponential tail for a large fixed C, using the node total and v_1\ge r_b.

The identity in Lemma 2.9, integrated using the nonnegative bank Z, gives

The geometric errors are absorbed by part of the remaining margin \eta\nu/4: on |u|\le T they carry the factor Ce^{-\kappa T}(1+T^2)=o(\eta), and the tails are negligible under (4.24) and the node total. The nonnegative C_Q\Lambda term has been discarded.

There is a fixed c_1>0 such that A(v_1)\ge c_1I_1. This includes a current hit with q_{v_1}>\lambda_{v_1}: although its current test is omitted, the next larger mesh scale t_+\le2v_1 obeys \tau_{t_+}\le2E_{2i+1}(t_+), and G_{v_1}\le(t_+/v_1)^mG_{t_+} bounds its current average. Only q_{v_1}\ge1 is needed for the lower bound.

For an explicit cumulative-sign argument, fix C_Q\ge1 first and let C_s be the constant in (4.26). Write E_1=E_{2i+1}(v_1) and take T large enough that its tail is at most E_1/2 and c(T)\le1/2. Then

If A(v_1)\le c_1I_1 with c_1\le\alpha/8, then L'_{v_1}\le2c_1E_1-(D+\Lambda)_{v_1}/2\le-\beta E_1, where \beta=\alpha/4. For v\in[v_1,2^{1/4}v_1], differentiating the fixed source L' and dominating |\dot G_v| by CG_{2v} gives |\dot L'_v|\le K_1E_1. The constant K_1 is fixed after C_Q; the envelope at 2v follows from strictly larger mesh tests, and v_1<\rho/16 leaves room. In particular this estimate does not use the omitted current test or differentiate the fixed far cutoff. Choose

On [v_1,e^\theta v_1] one has L'_v\le-\beta E_1/2, hence \dot Z\le-\beta I_1/2. Nonnegativity of Z at the right endpoint gives Z(v_1)\ge\gamma I_1. Finally take T large enough that the error in (4.28) is at most \gamma I_1/2. That inequality at v_1 yields A(v_1)\ge\gamma I_1/2>c_1I_1, a contradiction. All constants were fixed before this last largeness threshold; only the cumulative sign of Z was used at later scales.

The upper differential inequality \dot A\le2A+v^2R_v+ absorbed errors, again from Lemma 2.9, and v_1/s_0\le(\log T)^{20} now yield

One may retain the geometric error explicitly in the integrating factor formula: divided by I_1 it is O(e^{-\kappa T}T^C/\varepsilon), hence smaller than every fixed inverse power of \log T. Inserting (4.29) in (4.28) at \rho gives A(\rho)\ge c(\log T)^{-40}I_1. On the other hand the node mass gives

The ratio of the upper bound to I_1 is at most CT^{-J_0+4N(m-2)}, contradicting (4.13). This excludes current hits and all side equalities within the stated logarithmic factor.

Gaussian children and the bottom. For any remaining contact, configurations on the Gaussian cells at s_0 match the node configuration with movement at most e^b\sqrt d. Here is the comparison chain for a cell with work radius comparable to s_0/U, where U=1+|y-x|/s_0\le C\sqrt T. First shrink concentrically at x_* to a small fixed multiple of R, move the center to x at that radius, and then shrink concentrically to a work radius comparable to Us_0. Move the center to y at that radius and finally shrink to the cell radius. At each stage change radii by factors at most two and make center steps at most one quarter of the smaller inner majority radius. All these inner balls satisfy (4.9), so (4.10) bounds the changes by the sum of their local matching errors. The initial fixed chain stays inside B_{4R}(x_*); the off-center chain at scale s_0 lies in B_{CUs_0}(x)\subset B_{2R}(x_*). The envelope bounds these increments by (4.16), with fixed polynomial losses. More explicitly, above \rho the node total gives, for \rho\le t\le cR,

Off-center balls nested from scale Us_0 down to s_0/U have the same increment bound as s_0\sqrt{E_{2i+1}(s_0)}, with another fixed power of U\le C\sqrt T, by the raw estimates proving (4.27). All normalized bounds used for intermediate configurations are at most a fixed power of T times E_{2i}(R) or E_{2i+1}(t) for t\ge cr_b/T^2. Hence they tend to zero by (4.15). Their |M| terms give the same smooth mass test selecting Q, and Lemma 2.7 is applied with the fixed \delta_{\rm maj} before each matching. The logarithms of all these factors are O_{J_0,N,m}((\log T)^2)+O((\log T)^{60}) after division by \sqrt d. Since there are O(T) steps, (4.16) proves the claimed e^b\sqrt d allowance.

Here is the precise normalization of the cells. A smooth partition of unity on |u|\le C\sqrt T, where u=(y-x)/s_0, can be chosen with bounded overlap on balls B_{r_j}(y_j), with

Use a sufficiently small fixed constant in \asymp to leave room for B_{4r_j}(y_j). This follows, for example, by a maximal disjoint selection for the Lipschitz radius function in (4.30), followed by normalized smooth bumps. On the enlarged ball, G_{s_0}(x-y) differs from G_{s_0}(x-y_j) by a fixed factor, and its logarithmic gradient is O(r_j^{-1}). Thus

is an admissible child weight after a fixed normalization. In particular, this normalization has no power of T.

The child’s unweighted density was estimated in (4.27). The identity

then gives

by the choice of J_0, with C independent of i,T. Also r_j\ge cr_b/\sqrt T\ge cr_b/T^2. The smooth mass test following (4.27) identifies Q on each child. After scaling its work ball to one, choose from Lemma 2.7 a majority configuration on B_{r_{\rm maj}} with local upper bound C r_{\rm maj}^{-m}E_{2i+2}(r_j). Using the same normal frame, multiply its heights by r_j to return to the root coordinates. The matching argument above gives its allowance relative to \mathcal A, so these are admissible next-generation nodes. Although the raw density estimate loses a polynomial factor, summation of the resulting estimates does not: bounded overlap and Gaussian comparability give

The fixed d floors therefore sum to the correct constant-density Gaussian average.

At a bottom contact s_0=r_b, these cells are small nodes. The stationary-piece argument above, including (4.22), bounds L'_{s_0} by o(\eta)\nu_{s_0}. The Gaussian tail outside C\sqrt T is o(\eta)d for a sufficiently large fixed C: combine its e^{-C^2T/4} factor with total node mass and s_0^{-m}\le\exp(O(T)). This contradicts (4.7); there are no bottom contacts. At any other remaining contact the next-generation bound and (4.31) give

All omitted tails are absorbed by the subtractive margin in (4.7). In particular, every remaining contact is a vertical-side contact with some equality scale

First charging step: replace a Gaussian by a nearby ball. Put J=\log T and p=p_{i+1}. Call a contact (x,s) good if some closed ball of radius 0<r\le Js satisfies

where A_0\ge2C_0 is a fixed constant. Otherwise call it bad. We first show that all bad centers together carry only o(\eta)\mathcal N_R mass of h.

For the Euclidean Gaussian, radial layer cake gives

At a bad contact the failure of (4.34) for every r\le Js bounds the part with t\le J by h_s/(A_0p). Define the remaining layer-cake tail by

This is a tail in the layer-cake radius: it includes the residual weight c_ms^{-m}e^{-J^2/2}\nu(B_{Js}(x)) as well as mass farther away. Combining \nu_s\le h_s/(A_0p)+\mathcal T_{J,s}\nu with (4.32) gives

Let E_{\rm bad} consist of spatial contact centers that admit no good representative; for each of them assign one contact scale. The elementary 5r covering lemma selects disjoint assigned balls whose fivefold enlargements cover these centers. Their scales lie in [r_b,\rho], hence in O(T) dyadic bins. In a bin [a,2a), disjointness implies, for every y and t\ge1,

the disjoint balls of radius at least a have centers in a ball of radius Cta. Sum (4.35), use Tonelli, and then sum the bins to obtain

Here h^* denotes outer measure, so no measurable choice of a contact above each center is needed. The same proof on the torus uses periodic lifts of h,\nu: the small selected balls have disjoint lifts, the packing estimate is unchanged, and the periodic Gaussian is exactly the sum over those lifts. Alternatively the nonnearest copies can be absorbed in the exponentially smaller chart tail.

Second charging step: localize the union of good balls. Every good ball lies in a set determined only by the node’s tests and thresholds. To see this, let (x,s) be its contact and choose the equality scale v_1 in (4.33). For every mesh scale v\ge v_1 and every |y-x|\le Js, Gaussian differentiation gives

Indeed |\nabla G_v(y-z)| is bounded by Cv^{-1}G_{2v}(x-z) when |y-x|\le v/10; the envelope at 2v proves the assertion when 2v\le\rho. For 2v>\rho the node total and the top slack following (4.23) give the same bound. Since Js/v_1<J^{-19}, for sufficiently large T every point of the good ball satisfies

Order the mesh scales v_1',\ldots,v_K' from largest to smallest, and define the Borel set

Equation (4.38) says that every original good ball, not just its center, lies in \mathcal E_\lambda.

We use the following form of the Besicovitch covering theorem [19, Chapter 1, Lemma 4.5]. For a bounded set with one assigned positive-radius ball at each point and bounded assigned radii, there is a countable subfamily covering the set by the original balls, with multiplicity bounded by a dimensional constant N_m. Arbitrarily small assigned radii are allowed. One proof groups the radii into dyadic bins and, in decreasing bins, greedily selects uncovered centers. The selected centers in each bin are separated by that bin’s minimum radius, so only finitely many are needed in a bounded region. For the overlap bound at a point, divide directions into finitely many cones of angular diameter less than \pi/3. In a cone take the first selected center x_i among balls containing the point y. For a later such center x_j, the cosine law gives |x_i-x_j|<\max(|x_i-y|,|x_j-y|)\le\max(r_i,r_j). Since x_j was not covered by the earlier ball, |x_i-x_j|>r_i; hence r_j>r_i. The decreasing order of the dyadic bins forces these two radii into the same bin. Packing in that bin bounds the number. A selected center equal to y contributes at most one additional ball. This proves the dimensional multiplicity bound.

Choose one good contact and one good ball above each center that has such a representative, and apply this covering fact. Since the original balls lie in \mathcal E_\lambda, (4.34) gives

This is the reason for using original balls in this second covering; the inequality (4.34) need not hold on an enlargement.

Choosing the thresholds. The measure \nu and the functions q_v are already fixed by the node. Choose independent thresholds at the mesh scales with distribution function

For every fixed K_0>0, uniformly in real z,

If F(z+\delta)\le T^{-K_0} this is immediate. Otherwise, put a_*=(1+K_0\log T)^{-1}. One has z+\delta-1\ge a_*, also when z+\delta\ge2, and consequently z-\delta-1\ge a_*-2\delta\ge a_*/2 for large T. Thus this case does not meet the lower endpoint 1. On the part of [z-\delta,z+\delta] below 2 one has (\log F)'=(t-1)^{-2}\le4/a_*^2\le C_{K_0}(\log T)^2; above 2, \log F=0. Integration proves F(z+\delta)-F(z-\delta) \le C_{K_0}(\log T)^{-13}F(z-\delta). Truncation at 2 only decreases this difference. This proves (4.42), including intervals meeting either endpoint.

Fix y, and write z_k=q_{v_k'}(y) and r_k=F(z_k-\delta). At the kth event in (4.39), all preceding thresholds have survived, namely \lambda_{v_j'}\ge z_j-\delta. Independence, (4.42), and a union bound give

The sum telescopes to 1-\prod_k(1-r_k)\le1; no disjointness of the candidate events has been assumed. Since K=O(T), any fixed K_0>2 suffices. Integrating against the fixed measure \nu and using Tonelli supplies thresholds with

Choose such thresholds first, and only then construct the contact measure. This order makes independence explicit: the contact measure may depend on the chosen thresholds, but neither \nu nor q_v does. For large T, the last bound pays for A_0,N_m and the fixed factor between \nu(X) and \mathcal N_R. Equations (4.36) and (4.40) imply

Because h(X)=\int w\,\dd L'-\eta\nu(X), this is (4.18) after increasing the fixed C.

Finally solve the recurrence:

At the initial node \mathcal N_1\le2d. Increasing the fixed N relative to the requested P gives (4.12). There have been at most n_0 configuration changes, costing (n_0+1)e^b\sqrt d\ll S(T)\sqrt d in total. Thus the fixed root separation and far cutoff used at every step are valid. A polynomial weight or dilation loss is handled by starting with a correspondingly larger P and using the same Gaussian cell normalization. This proves the final assertion as well.□

Proof. Only the first assertion needs proof. Put

and let \mathcal C_\sigma be the centers of contacts in K_\sigma\setminus\mathcal G_\sigma. Assign these contacts to grid cubes of side \sigma. If such a cube J contains a contact x_J, the Gaussian lower bound on J, together with the contact inequality, gives

Since \beta_\sigma\le h_\sigma, this also bounds \beta_\sigma(J). On an occupied half-open grid cube define

and put g_\sigma=0 on unoccupied cubes. Then \overline\beta_\sigma\le g_\sigma\,\dd y and 0\le g_\sigma\le Cd_*. For every continuous test \varphi,

where \omega_\varphi is its modulus of continuity.

For a finite measure \lambda, at almost every Lebesgue point of its absolutely continuous density one has

For the singular part the corresponding limit is zero almost everywhere. To check the assertion, decompose into dyadic annuli about y; the differentiation bound on the mass in each sufficiently small ball is uniform in the indicated displacement, and the remaining annuli have an exponentially small Gaussian weight. Apply this first to smooth approximations of the absolutely continuous part and then to its L^1 remainder. Apply this assertion simultaneously to \tau and f. If \tau_{\mathrm{ac}}(y)>d_*, the cube containing y is eventually unoccupied, since all its contacts satisfy \tau_\sigma\le d_*. At every other simultaneous differentiation point the uniform displacement bound controls the supremum defining g_\sigma. Therefore

Reverse Fatou, using the common majorant Cd_* on the fixed chart, and the weak-test estimate above show that every bottom limit has density bounded by

The value of the fixed constant in the threshold is immaterial.

At a point in this set, the area formula and Lemma 2.3 give exactly Q projection entries, counted with their integer multiplicities. Indeed the difference between their integer count and Q is bounded by C\tau_{\mathrm{ac}}<1. Every entry has e\le Cd_*; an entry with larger tilt would itself contribute more to the projected density of D. The part on which the projection has deficient rank projects to a Lebesgue null set, by the area formula for rectifiable sets. The expansion of the projection Jacobian at each remaining entry is

Sum this inequality with weight \theta/J_\pi. Since the count is Q, its left side is exactly the density of L, with respect to base volume. Taking C_Q large enough absorbs the displayed geometry term and gives (f_{\mathrm{ac}})_+\le Cd_*D_{\mathrm{ac}} on the set in (5.4). Its integral is at most Cd_*d. Compactness and (5.4) prove the stated uniform limsup bound, hence the assertion.□

Proof. Write I_{\mathrm{end}} for the artificial term in (5.7). The definition makes I positive and continuous. Let

We apply the following argument first at s=s_0, and then after initialization up to the first crossing of a=T/2. At s_0 the threshold and K>2 give the exact value J_\tau(s_0)=s_0^2d_*, realized at v=s_0, whereas I_{\mathrm{end}}(s_0)=o(s_0^2d_*). At any later scale before that crossing, I(s)/s^2\ge e^{-T/2} and I_{\mathrm{end}}(s)/s^2\le e^{-T+o(T)}, so I is realized by an actual future value of A. Since \tau\ge|M|, J_\tau\ge I>I_{\mathrm{end}}. At a maximizing scale v, writing \delta=\tau_v, these observations give

The first lower bound holds at initialization as well. For any fixed N, a maximizer with v\ge\rho/T^N would, by the root total mass bound, satisfy

which is impossible. Thus all fixed polynomial dilations needed below, including the body C\sqrt{|\log\delta|}\,v, lie within \rho.

Here is the raw density estimate on that body. A cell centered at y, with |y-x|\le Rv and radius r=c v/(1+R), lies with its fixed enlargement inside a ball of radius C(1+R)v about x. The Gaussian lower bound at that radius and (5.11) imply

For R\le C\sqrt{|\log\delta|} this is small, uniformly in \delta\le d_*. Moreover \tau includes |M|=|\pi_\#\mu-Q\vol_S|, so the normalized smooth mass average on each cell is close to that same Q. The integer in Lemma 2.5 is therefore Q on every cell. The Gaussian has a bounded ratio on a fixed enlargement of such a cell, and Corollary 2.8 sums the estimates with bounded overlap.

Annular summation of (5.11) bounds the Gaussian tail beyond Rv, including any fixed derivative polynomial, by C\delta\operatorname{poly}(R)e^{-cR^2} up to \rho. Above \rho the root mass bound gives a remainder at most Cd v^{-m}\operatorname{poly}(\rho/v)e^{-c(\rho/v)^2}. Since v<\rho/T^N, \delta\ge e^{-T/2}, and |\log\rho|=o(T), this is o(\delta), uniformly even as v tends to zero. Increasing the fixed body constant makes the full tail o(\delta). Consequently

Suppose A(v)\le\varepsilon v^2\tau_v. For C_Q sufficiently large and then \varepsilon sufficiently small,

Gaussian differentiation and the maximizing dilation bound show that the derivative, in logarithmic scale, of 2M_r-D_r-C_Q\Lambda_r is bounded by C_K\tau_v on a fixed short outward interval. Thus, for a fixed \vartheta>0, (f_*)_r\le-c\tau_v/2 on [v,e^\vartheta v]. The negative margin may simply be omitted in obtaining this upper bound. By (5.6),

if \varepsilon<c'\vartheta. This contradicts the cumulative contact sign. Hence A(v)\ge c v^2\tau_v at an uncontrolled maximizer. At s_0 this first proves I(s_0)\ge c s_0^2d_*; at each subsequent path scale it proves the last inequality in (5.8). This order uses only maximality and the root mass bound when establishing the raw cell estimates. The same cell argument, now using the proved dilation bound, gives (5.10) at every path scale; any prescribed inverse power is obtained by enlarging the Gaussian body constant.

At s_0, the threshold equality and the just proved bound give I(s_0)\ge c s_0^2d_*. The opposite bound follows from A(v)\le v^2d_*, K>2, and the negligible artificial endpoint term. For that last point, d_*\le Cd s_0^{-m} implies s_0\le C\exp(-(T-a_0)/m)\ll\rho. At the outer endpoint the artificial term dominates, and the exact formula is

Before the first crossing of T/2, the nonendpoint terms have I(s)/s^2\le d_*, and the endpoint term has I_{\mathrm{end}}(s)/s^2\le d\rho^{-2m-4}=o(d_*). This proves the remaining assertions about the levels.

On any compact logarithmic scale interval, A is smooth and has bounded derivative. The supremum of v^{-K}A(v) over [s,\rho], with the positive endpoint floor, is therefore locally Lipschitz. Its derivative vanishes unless its maximum is realized at the moving endpoint v=s. On the equality set, the derivatives of I and A agree almost everywhere. This proves the reset rule. Gaussian differentiation, the dilation bound, and Lemma 2.9 give |\dot A|+|\dot Z|+|\ddot Z|\le CI. The constant d term is negligible, since s^2\eta d/I=\eta d e^a\le e^{-T/2}. Together with the reset rule these prove (5.9) and the assertion about \dot Z/I.

Finally all polynomial dilations in a used here lie well inside the chart. Indeed I(s)\le Cd s^{2-m}+I_{\mathrm{end}}(s), whereas I(s)\ge s^2e^{-T/2} on the path. The endpoint term cannot account for this inequality. Thus s\le Ce^{-T/(2m)}, up to an immaterial fixed weakening of the exponent. This proves the claimed interior room and justifies the uniform Gaussian cutoffs.□

Proof. The bound on s_0 in the envelope proof gives s_0\operatorname{poly}(a_0)\ll\rho. Threshold control on fixed enlargements therefore selects Q by Lemma 2.5. Choose a joint majority-and-moment witness \mathcal A_0=\{a_1,\ldots,a_Q\} from Lemma 2.7 at scale s_0, using B_{r_{\rm maj}s_0}(x) and the fixed majority proportion when Q>1, and a fixed concentric inner ball when Q=1. Use its distinct entries as the initial wells. Put R_0=C_b\sqrt{a_0}, with a sufficiently large fixed C_b>2. All fixed enlargements needed on |u|\le R_0, where u=(y-x)/s_0, lie below \rho. Threshold control gives normalized density at most Cd_* on these dilations. The dilation part of Corollary 2.8, applied only on this body, and \Gamma_{s_0}\le C\dist(z,\mathcal A_0)^2 give

The root mass identity and threshold control at \sqrt2s_0<\rho give

The last term accounts for the fixed exterior-chart and periodic tails. Since \Gamma_{s_0}\le Cs_0^2, Gaussian comparison yields

For the periodized kernel the split and comparison are made termwise in its lifts. Consequently U(s_0)\le Cs_0^2d_*=O(I(s_0)). If the contact itself is a threshold equality, then Z(s_0)=0, which gives the first initialization, with one well when Q=1.

Assume now that \underline s<s_0. For \underline s\le v\le s_0, put R_y=|y-x|/v and consider the Gaussian work cell B_r(y) with r=cv/(1+R_y), where R_y\le C\sqrt{a_0}. For any fixed required enlargement B_{Lr}, the Gaussian at C_L(1+R_y)v is bounded below by a fixed multiple of ((1+R_y)v)^{-m} on that enlargement. This scale lies in [\underline s,\rho], so the threshold bound gives, in particular,

The same estimate holds for B_{Lr} after changing its fixed constant. Since d_c=o(1), it supplies small D+\Lambda energy and bounded normalized projected mass. Its |M| part puts the normalized smooth mass average within Cd_c<1/4 of Q, so Lemma 2.5 selects that integer on every cell. The cell radius may be below \underline s; the threshold is used only at the larger Gaussian scale.

When Q>1, scale B_r(y) to one and choose a length-Q majority list \mathcal A_{v,y} on B_{r_{\rm maj}} with local upper bound r_{\rm maj}^{-m}d_c and the fixed \delta_{\rm maj},C_{\rm maj}, as required by Proposition 4.3. Multiply its heights by r when comparing them in the fixed coordinates of this section. The overlap and matching rules (4.9)–(4.10) match these lists to \mathcal A_0. Along concentric dyadic scales from s_0 to v, the increments Cr'\sqrt{d_*} at radius r' sum to Cs_0\sqrt{d_*}. At the fixed scale v, enlarge to radius C(1+R_y)v, move the center in the prescribed steps, and shrink to r. All fixed enlargements in this part lie inside B_{C'(1+R_y)v}(x); at an intermediate radius r' their normalized densities are at most C((1+R_y)v/r')^m d_*. The matching increments therefore add only C(1+R_y)^C v\sqrt{d_*}. The total motion is bounded by a fixed polynomial in a_0 times s_0\sqrt{d_*}, uniformly even as the bottom tends to zero.

Suppose \mathcal A_0 has at least two nonempty groups separated by more than S_h(a_0)s_0\sqrt{d_*}. Group the Q entries of each \mathcal A_{v,y} according to the corresponding separated groups of the initial wells. Matching preserves their nonemptiness, and their gaps exceed 10S(T_c)r\sqrt{d_c}: the fixed power S_h(a_0) absorbs S(T_c), the displayed cell factor, and the matching loss, since T_c=a_0-O(\log a_0). These length-Q lists are the inputs to Proposition 4.3. The cell tests obtained by partitioning the nonnegative Gaussian are admissible after normalization in the cell coordinates. Bounded overlap and a sufficiently large input exponent absorb their fixed polynomial losses. In the curved case the physical cell radius is lr, and T_c=o(T), so the original physical smallness implies the cell hypothesis. The threshold and root tail bounds from the envelope proof apply with controlling density d_*, uniformly down to \underline s. Under nonnegative testing, f_*\le L-C_Q\Lambda, and the far term in Proposition 4.3 is nonnegative. The resulting bound, integrated in scale, gives

Entries too close to remain separate are merged immediately by the rule below. Their moment before merging is O(I(s_0)).

If there are no such separated groups, a chain through at most Q entries of \mathcal A_0 shows that its diameter is at most CQS_h(a_0)s_0\sqrt{d_*}. Use one common well. Pointwise, the new clipped square is bounded by a fixed multiple of the old one plus the squared diameter. The preceding U-bound and bounded Gaussian mass therefore give the stated estimate for U. It remains to check the surplus of I over Z. If A(s_0) is a sufficiently small fraction of I(s_0), this is immediate from (5.6). Otherwise A(s_0)\ge cI(s_0). When \underline s\ge s_0/2, the inequality \dot A\le2A+s^2R_s+\text{error} gives

These terms are retained in A(s_0)-Z(s_0), after absorbing the error into the positive margin.

When \underline s<s_0/2, Gaussian differentiation and (5.3) give |\dot A(v)|\le Cv^2\tau_{2v}\le Cv^2d_* on [s_0/2,s_0]. Since A(s_0)\ge cI(s_0)\asymp s_0^2d_*, for some fixed 0<\vartheta_0<\log2 one has A(v)\ge cI(s_0) throughout [e^{-\vartheta_0}s_0,s_0]. Apply Corollary 2.8 directly on this earlier interval, using the scalar hypotheses verified in (5.16). The tail implication in the envelope proof applies with controlling density d_*: here v<\rho/T^N for every fixed N, and d_*\ge e^{-T/2}. It gives an o(d_*) remainder uniformly in the bottom. Thus |M|_v\le C(D_v+\Lambda_v)+o(d_*); because M_v=A(v)/v^2\ge cd_*, this yields D_v+\Lambda_v\ge cd_*.

The surplus uses further information. Where \Lambda_v\ge cd_*, the retained term C_Qv^2\Lambda_v in J(v) pays it directly. On the complementary portion D_v\ge cd_*. Let V_v be displacement from the common well, smoothly clipped at distance comparable to v. Apply the moment part of Corollary 2.8 to joint configuration witnesses on the admissible dilations and match their centers to the common well. Gaussian summation absorbs their polynomial losses and gives

The support estimates for those chosen witnesses put all support heights on |y-x|\le C_b\sqrt{a_0}v within

of the common well. Clipping is therefore absent on this body. For the clipped tail, \pi_\#\mu=Q\vol_S+M controls the bare mass as well as the defect. Since \log S_h(a_0)=o(a_0), taking the fixed C_b sufficiently large makes the Gaussian mass tail o(d_*S_h(a_0)^{-2}). The defect and chart tails obey the preceding uniform bounds.

Choose one fixed root-chart cutoff 0\le\chi\le1, equal to one on the fixed working region, whose transition is at distance at least h from that region and has width at least ch, decreasing h to the available width. Then \|\nabla\chi\|_\infty\le C(1+h^{-1}); the first-variation product rule differentiates \chi once, while all scale derivatives leave it fixed. With u=(y-x)/v, normal stationarity in Lemma 2.10 now gives

The factor v^{-1} is the Gaussian derivative. Retaining |V_v| in the curved normal error and using its moment bound leaves relative error e^{-\kappa T}\operatorname{poly}(a_0,S_h(a_0))=o(1); the normal identity introduces no further inverse power of v. Thus the flux has size at least cvd_* on the D-portion. Cauchy–Schwarz and |P_{vh}u|^2\le B[u,u] yield

After the margin absorbs the errors, \dot A-\dot Z retains v^2R_v+C_Qv^2\Lambda_v; at all other scales it is nonnegative by (5.6). Integration over the fixed logarithmic interval therefore proves the surplus claimed in the statement. Since the one-well coefficient \alpha below is less than one, k_0=cS_h(a_0)^{-6} pays for the moment while leaving a strict initial inequality. The same proof for Q=1 uses the Gaussian-summed moment with fixed S_h and no separation step.□

Proof. We specify the mergers first. At stage j, merge a closest pair as soon as its gap g satisfies

Retain one of the two centers. Take \gamma>0 sufficiently small, C_{j+1} a sufficiently large fixed multiple of QC_j, and replace k by

Perform further mergers immediately if required. For Q=1 there is no merger rule. A common-well initialization when Q>1 may be counted as an initial merger; it uses no more than the available Q-1 losses.

The solution of (5.17) on a stage starting at level a_j is

Retracing levels therefore causes no accumulated loss. Since a\ge a_0-O(1), k changes by at most a fixed factor on a stage. There are at most Q-1 mergers, and their levels are at most T/2. Thus

where Lemma 5.5 verifies the final comparison. In particular e^{-a}/k is exponentially small uniformly along the path.

We next verify that the wells continue to represent the configuration. At a scale s, let \mathcal A_s be a length-Q majority list on a fixed inner ball, using the fixed convention above when Q>1. For support control, choose joint majority-and-support witnesses from Lemma 2.7 on the balls under consideration and match their centers to these lists. The support and matching bounds, together with (5.8), bound configuration changes on a ball of scale v, including polynomial enlargements, by Cv e^{-\beta a(v)} for some fixed \beta>0. Decrease \beta if necessary so that \beta|\dot a|\le1/2. Summing on a geometric sequence from s_0 to s gives Cs e^{-\beta a(s)}. The same estimate tracks each initial configuration entry. The additional displacement caused by previous mergers is bounded by the sum of their gaps. Choose \gamma\ll\beta and \gamma|\dot a|\le1/2. Then s e^{-\gamma a(s)} increases outwards, and the successive enlargements of C_j make all earlier capped gaps a small fixed fraction of the current cap. These facts give, in particular,

for some fixed \gamma'>0. This estimate does not use the barrier. It also bounds the distance from retained wells to the corresponding tracked configurations. The common-well initialization has the same property, since \log S_h(a_0)=o(a_0).

While the barrier holds, each entry of \mathcal A_s is matched on a fixed majority of the inner ball to a projection entry within Cs e^{-a/2}. The area formula and the uniform upper bound for J_\pi give that entry a fixed amount of Gaussian mass. The bound |D\Gamma_s|^2\le C\Gamma_s and the numerical barrier therefore give

Since \Gamma_s is comparable to \min\{\dist(z,\mathcal W)^2,s^2\} and e^{-a}/k\to0, every entry lies within Cs\sqrt{e^{-a}/k} of a well. This distance is a small fraction of S_1(a)se^{-a/2}k^{-1}. It is also a small fraction of the cap, using \log(1/k)=o(a_0); alternatively (5.23) gives that conclusion directly. When no merger is triggered, the resulting neighborhoods of the wells are disjoint. The group represented by a well is the sublist of \mathcal A_s in its neighborhood. When Q>1, its integer count cannot change: on two sufficiently nearby scales the majority balls obey (4.9), their lists match by (4.10), and the matching error is much smaller than the gap. For Q=1, the sole length-one group remains represented. At initialization each well is represented. At a merger the surviving neighborhood contains the entries of both merged wells after any further immediate mergers. This proves representation at every stage, including a first barrier equality.

With several wells, Q>1, so we may apply Proposition 4.3. For a Gaussian cell with R_y=|y-x|/s\le C\sqrt a and r=cs/(1+R_y), the dilation bound at C(1+R_y)s gives

The cell and its fixed enlargements lie in the chart, and T_c=a-O(\log a)\to\infty. As in (5.16), the |M| bound selects Q. In the cell coordinates choose a length-Q majority list \mathcal A_{s,y} on B_{r_{\rm maj}}, with local upper bound r_{\rm maj}^{-m}d_c and the fixed \delta_{\rm maj},C_{\rm maj}, and multiply its heights by r for comparison in the section coordinates. The off-center overlap chain from initialization now uses (5.8); matching to \mathcal A_s costs at most a fixed polynomial in a times s e^{-a/2}.

Group the Q entries of \mathcal A_{s,y} by the represented wells to which their matches in \mathcal A_s belong. The wells index these nonempty groups; the input list still has length Q after mergers. Before a merger, if the second threshold in (5.21) is smaller, the gap is larger than S_1(a)s e^{-a/2}k^{-1}. This dominates the entry-to-well and cell matching errors and 10S(T_c)r\sqrt{d_c}, by the choice of S_1. If the cap is smaller, the ratio of its gap to the last quantity tends to infinity, since it is bounded below by \exp((1/2-\gamma)a-O(b(a))-O(\log a)). Thus the groups meet the separation premise in both regimes. Away from the clipping tails, every point of the transition set is a fixed fraction of a gap from the represented groups, and remains in the cell’s far region after matching. This includes every extra annulus in D_{\mathrm{tr}}.

Apply Proposition 4.3 to the nonnegative normalized cell weights. Their polynomial Lipschitz costs are absorbed by taking its exponent larger than P_0, while c(T_c)\asymp(\log a)^2/a. Bounded overlap and the Gaussian tails give total remainder O(a^{-P_0}I/s^2). The curved hypothesis is inherited because the physical cell radius is lr and T_c\le T/2+O(1). This proves (5.20).

We prove the two differential inequalities explicitly. In flat coordinates the horizontal test with multiplier \Gamma_s gives

Normal stationarity with G_sD\Gamma_s converts the cross term to s^2\int G_sD^2\Gamma_s:P_{vv}\,\dd\mu. On the pure regions D^2\Gamma_s=2\operatorname{Id}, and elsewhere its norm is bounded. Dropping -\Gamma_se, and using (5.23) to bound the \Gamma_s B[u,u] term, gives (5.18). The explicit derivative \dot\Gamma_s=2s^2[\chi(t)-t\chi'(t)], t=\Gamma_0/s^2, is supported on the clipping region. That region does not meet the Gaussian body by (5.23).

For the second inequality use |P_{vh}u|^2\le B[u,u] and complete the square:

Normal stationarity gives

up to the already specified tails. This proves (5.19) in flat coordinates.

For completeness the tails here also include terms involving mass without a tilt factor. Since \Gamma_s\le Cs^2, the mass tail beyond C\sqrt a is bounded by Cs^2e^{-cC^2a}, plus the dilation-controlled defect. Taking C large makes this smaller than any required multiple of a^{-P}I. Terms with tilt use (5.8) directly. No inverse power of k is required for these flat errors, because the derivative bounds in (5.14) are uniform. For the curved errors, keep the original coordinates scaled by the physical radius l, and write \langle f\rangle_s=\int G_sf\,\dd\mu and \gamma_{\rm geom}=l+\sup|z_{\mathrm{phys}}|. The dilation bound gives \langle(1+|u|^2)(e+\lambda)\rangle_s\le P(a)I/s^2, where P is a fixed polynomial and \lambda=|z_{\mathrm{phys}}|^2+|H_S|^2+|\nabla H_S|^2. For either v=V_b or v=D\Gamma_s, \langle|v|^2\rangle_s\le CU\le CI/k and |Dv|\le C. After multiplying the normal identity by s^2, the errors in Lemma 2.10 satisfy

Here e\le m controls the term |v|e; the last line is the error from differentiating the Gaussian. In the horizontal test, \Gamma_s\le Cs^2 bounds the error carrying \Gamma_s(e+\lambda) by C\gamma_{\rm geom} P(a)I, while

Coordinate and base-volume comparison may also produce a term C\gamma_{\rm geom}\langle(1+|u|^2)\Gamma_s\rangle_s. Split at C\sqrt a; its body is bounded by C\gamma_{\rm geom} P(a)U, and its clipped tail has the already proved negligible bound. Thus it costs at most C\gamma_{\rm geom} P(a)k^{-1}I. These estimates retain physical heights and curvature inside \Lambda until after Cauchy–Schwarz; the displayed factors of s cancel the Gaussian inverse-scale factors. They give total error Ce^{-\kappa T}\operatorname{poly}(a,k^{-1})I in the moment inequality and this bound times k after square completion in the radial inequality. Since \log(1/k)=o(T), these errors are smaller than the stated \mathcal E I and \mathcal E kI. Chart-cutoff errors are still smaller because s\le e^{-cT}. This proves both differential inequalities with their asserted uniform errors.

It remains to prove the comparison, including its signs. Work at an equality kU+\alpha Z=I on a fixed stage and put

Differentiating I-kU-\alpha Z, and using \dot a=2-\dot I/I, gives

At a reset write \mathcal D=s^2D_s, \mathcal R=s^2R_s. Apart from the absorbed geometric error,

Insert (5.18) and (5.19) into (5.25) and use kU=I-\alpha Z. The resulting lower bound is

In both cases h_1=o(1), so c_J>0; discarding the margin therefore has the indicated favorable sign. The first-variation error in \dot A, which initially need not contain k, is absorbed by a fixed fraction of this positive margin. Equivalently, its remaining contribution divided by kI can be included in \mathcal E, since \log(1/k)=o(T).

With one well,

Thus the \dot Z coefficient cancels exactly at Z=0. The estimate |\dot Z|\le CI bounds its remaining contribution by O(k/a^2)Z. The term (4k-2)qI is at least ckI/a^2; the coefficient of Z left in the first two terms of (5.26) is at least ck. Hence the reset derivative is strictly positive for large a.

With several wells there is the useful exact cancellation

In particular \delta_\alpha=o(k), |h_1|\le C/a, and

Multiply (5.20) by this negative coefficient. It supplies a favorable multiple of C_1k\log a\,s^2D_{\mathrm{tr}}, which absorbs the transition loss in (5.26). Its adverse error is O(k a^{1-P_0}/\log a)I=o(kI/a^2). The remaining Z terms are positive, since their possible loss is O(\delta_\alpha)Z=o(k)Z. Again (4k-2)qI\ge ckI/a^2 supplies strict positivity.

Off the reset set \dot I=KI, and \mathcal D\le2I-\dot Z. The coarser consequence \dot U\le C\mathcal D+\mathcal E I gives

The coefficient Ck-\alpha is negative. A negative \dot Z therefore helps. For one well, \dot Z\le2I; for several wells, (5.20) gives \dot Z\le a^{-P_0}I, and \alpha a^{-P_0}=o(1). Taking K large and the initial upper bound for k small makes the derivative strictly positive. All the strict inequalities hold in a neighborhood of equality on each compact stage. The absolutely continuous comparison principle therefore applies despite the reset set having only almost-everywhere derivatives. Initialization is supplied by Lemma 5.3; in its multiple-well case choose P>3, so that the initially large coefficient \alpha still leaves strict room.

Finally consider a merger. Pointwise distance comparison and bounded Gaussian mass imply

Moreover \alpha_{\mathrm{new}}\le(1-ck)\alpha, whether several wells remain or the last two merge: in the first case use the formula linear in k. In the second case write \alpha=1+x, where x=C_1ka/\log a. The fixed upper bound for k and large a give x/(1+x)\ge ck, so \alpha_{\mathrm{new}}<1\le(1-ck)\alpha. Set \theta=Ck_{\mathrm{new}}/k and \lambda=Ck_{\mathrm{new}}S_1^2k^{-2}. By (5.22), \theta\ll1 and \lambda\ll k. The old comparison then gives

For the last inequality take the maximum over 0\le Z\le I/\alpha; both endpoint values are below I. Thus every merger preserves the barrier, completing its proof.□

Proof. We spell out the rate comparison to keep track of the inverse. Let Q\ge2 and j=2Q-2. Above the iterated-log threshold,

For b chosen as above, inversion of any fixed power of S_1(x)=e^{Cb(x)} first takes a logarithm, then applies F_{Q-1}, and then inverts (\log x)^3. In particular the inverse retains the outer exponential from this last step. For example when Q=2 its leading expression, up to fixed losses, is

It follows directly from (5.28) that, for some fixed \varsigma>0, every such inverse dominates

Any fixed number of inverse compositions, including fixed powers of the argument and value, retains this lower bound with a smaller positive \varsigma. To compare with F_Q(T)^C, take 2Q-1 logarithms: the lower bound has last value \exp(\varsigma\log_{2Q}T), whereas the latter has last value \exp(\sqrt{\log_{2Q}T}), up to lower order terms. The former is larger for every fixed C.

The same comparison shows b(T)=o(F_Q(T)^c) for each fixed c>0, and hence b(T)+b(a_0)=o(a_0), as used above. It also shows that after a fixed number of the inverse operations, \log B_1\gg b(a_0). This stronger comparison absorbs the initial factor S_h(a_0), including any fixed power of it. Enlarge C_*, if necessary, to exceed the finitely many powers arising from Q-1 applications of (5.22); proving the stronger separation also proves the originally requested one. Now choose B_Q to be a small fixed multiple of T, and choose the earlier bands successively by inverse operations with enough fixed slack to ensure (5.27). The comparisons just proved give all the lower-end requirements. For Q=1, F_1(T)=T^{o(1)}, so the claimed single band works and no inverse construction is required.

Consider the path from initialization to its first hit of each successive level 2B_i. There are Q resulting consecutive time intervals and fewer than Q merger events, counting a common-well initialization if needed. At least one interval has no event. On it take the last preceding hit of B_i before the first hit of 2B_i; this is a traversal staying in that band. All earlier mergers have levels at most 2B_{i-1} (or are initialization events). Iterating (5.22), with at most Q-1 steps, bounds their total loss by a fixed power of S_1(2B_{i-1}), the initial S_h(a_0), and fixed constants. Equation (5.27) and the explicit solution of (5.17) then give k\ge B_i^{-\eps_0} throughout the traversal. The remaining assertions are the estimates already proved. The number of wells only decreases, so a traversal with several wells has had several wells at every earlier stage.□

Proof. We first record precisely the differential information used here. By Lemma 5.2 and Proposition 5.4, 0\le Z\le A\le I, A'-Z'\ge0, and

almost everywhere. The last estimate follows also directly by differentiating the Gaussian source: its first two logarithmic derivatives are bounded by Gaussian averages of fixed polynomial weights times \tau, and the dilation estimate bounds these by CI/s^2. The constant background in the source is negligible since a\le T/2. At a reset I=A we have I'=A' almost everywhere on the reset set. Off that set I'=KI and a'=2-K<0. The weighted scalar estimate is

where P can be chosen as large as needed. With several wells,

Here D_{\mathrm{tr}}(s) is the Gaussian average over the enlarged transition set of Proposition 5.4. This set includes all regions where derivatives of either \Gamma or V_b differ from their near-well values, in particular the cutoff annuli of V_b inside the quadratic neighborhoods of \Gamma; clipping tails are estimated separately. These inequalities retain the cumulative contact condition Z\ge0; no sign for Z' at a later scale is asserted.

Here is a time-bin estimate that does not require monotonicity of a or of I. Fix an endpoint t_1 of an interval on which a\ge A_*>2, and put M_* =\max_{t\le t_1}I(t) on that interval. For nonnegative integers j and integers n\ge\lfloor A_*\rfloor, the simultaneous conditions

confine t to an interval of length at most one, because

The total length of the set of such times, even if it has many components, is therefore at most one. Summing the integral over these sets gives

For an interval confined to [B,2B], the same estimate has CB^{1-P} on the right.

Suppose first that the selected traversal has several wells. All earlier stages also had several wells, since the construction only removes wells. Its initialization gives Z(t_0)\le a_0^{-P}I(t_0). By (6.6) and (6.8), the positive variation of Z up to any endpoint is at most \epsilon M_*, where \epsilon\to0 as a_0\to\infty. Since A-Z is nondecreasing, for u\le t we have

The future envelope has the elementary property

Indeed a term realizing I(u) either comes from an A(v) with v\le t, when its exponential weight is at most one, or comes from a later v or from the artificial endpoint, when the same term at t is larger by e^{K(t-u)}. Approximating a supremum proves the assertion if it is not attained. Combining these facts, and taking a_0 large, yields

This argument is unchanged at a merger: I,A,Z do not jump, and (6.6) holds on both adjacent multiple-well stages.

Choose a traversal [t_b,t_e] with a(t_b)=B, a(t_e)=2B, and B\le a\le2B between them. Let M(t)=\max_{t_b\le v\le t}I(v) and I_b=I(t_b). On this interval

Fix a small constant \delta_1>0 and set h_B=\delta_1(\log B)^2/B. We bound the levels whose first increasing passage has Z'/I<-h_B.

Before M first reaches B^{10}I_b, split the range of M into successive factors of two. There are O(\log B) such pieces. Applying (6.8) on each piece shows that

Integration by parts against the continuous nondecreasing function M gives

Consequently \int (Z')_-/M\le2 for large B. In view of (6.10), the time spent with Z'/I<-h_B is at most C/h_B. By |a'|\le C, the image of these times in the level variable has length at most

Constants here may depend on the fixed \delta_1.

After this growth of M, (6.11) implies Z(t)\le C(B^{-10}+B^{1-P})I(t). If Z'/I<-h_B at a time with B+1<a<2B-1, (6.4) keeps Z'/I<-h_B/2 on a following interval of length c h_B. The interval stays in the traversal, and I changes by at most a fixed factor there. It would decrease Z by at least c h_B^2 I(t), exceeding the available value of Z(t) for P>12 and large B. This contradicts Z\ge0. Thus no such bad time occurs after the stated growth.

For completeness, first increasing passages can be selected without assuming that a itself is increasing. Use the running maximum of a starting at t_b. It is Lipschitz. Except for a null set of levels, a first passage is a differentiability point with a'>0: the image under a Lipschitz function of its nondifferentiability set is null, and the image of its zero-derivative set is null by the one-dimensional area formula. First passages so obtained are ordered. Since off resets a'=2-K<0, they are resets. The excluded levels are bounded by (6.12).

At every remaining passage I=A, so M_s=I/s^2=e^{-a}. The lower bound Z'/I\ge-h_B and the definition of the source imply

Apply (6.5). Taking C_Q sufficiently large compared with C_0, and then B large, forces D_s\ge c e^{-a}. Moreover (6.6) gives

Choose \delta_1 first so that the last bound is at most \vartheta e^{-a}. This proves (6.2) on all but o(B) of the first-passage levels in the band.

Now suppose there is one well. If no scale in a traversal of a level interval of length B^{1/4} satisfied (6.3), then throughout it s^2D_s<B^{-\eps_0}I. At a reset the Gaussian horizontal identity gives

Also (6.5), together with M_s=e^{-a}, forces \Lambda_s\ge c e^{-a}. Thus Z'\le-I when C_Q is sufficiently large. At a nonreset, the same scalar estimate and C_Q>2C_0 instead give

These estimates explicitly include the possible geometry contribution. In the flat case a reset with such small tilt is already excluded by the scalar estimate.

Put W=Z/I. It is nonnegative and initially at most one. At resets W'\le-1; at nonresets W'\le C B^{-\eps_0}. If the two time sets have lengths \mathcal R,\mathcal N, integration gives

On the other hand a'\le1 at resets, while a'=2-K at nonresets. A net increase of B^{1/4} therefore requires

a contradiction for large B.

In the multiple-well case select good first-passage levels greedily with separation 2B^{1/3}; removing an interval of length at most 2B^{1/3} per choice leaves at least cB^{2/3} choices. In the one-well case use ordered, disjoint level intervals of length B^{1/4} separated by 2B^{1/3}, and choose a witness in each corresponding first-passage traversal. These traversals are ordered in time. Discarding witnesses if necessary gives (6.1) and the asserted number.□

Proof. A finite family as stated can be constructed by using, in each coordinate, the three dyadic grids whose shifts alternate between the one-third positions at successive generations; this standard Euclidean construction is described in [13, p. 2]. At a scale between a fixed multiple of the ball diameter and twice that multiple, one of the three boundary sets in each coordinate is farther than the ball diameter from the center. Taking their product gives the required cube. We shall apply this property to a larger concentric ball, so the first inclusion in (6.13) has room to spare.

Let V_b be the normal multiplier of Proposition 5.4, and put u=(y-x)/s. In the scale-s height coordinate z'=z/s, define v(z')=V_b(sz')/s. Thus D_{z'}v(z')=DV_b(sz'), equal to the identity when sz' is near a represented well. In all integrals below, v means v(z/s)=V_b(z)/s, while the measures remain in the section coordinates. Dividing the normal first-variation identity in Proposition 5.4 by s^2, with its Gaussian cutoff, therefore gives

For multiple wells the discarded terms are bounded by a fixed multiple of D_{\mathrm{tr}}(s), and \vartheta in Lemma 6.1 is chosen small enough to absorb them. For one well there is no transition contribution. Clipping and chart tails have the arbitrarily small bounds in Proposition 5.4, and the curved first-variation errors are o(B^{-\eps_0}\delta). This last assertion follows from k\ge B^{-\eps_0}, the moment barrier, the dilation bound, and the physical exponential smallness in Theorem 3.1.

We give the moment and tail details needed to use this flux. Fix the witness scale s, and keep its wells, clipping scale, and multiplier V_b=V_b(s) fixed while the ball radius varies. Choose a fixed p>2 in the range (2.17), and fix an exceptional proportion small enough for the concentric dyadic comparisons below, independently of T, a, and R. For R\ge1 in any fixed polynomial range in a, the dilation bound and a Gaussian lower bound give

on the required fixed enlargement. Its |M| part identifies Q by the smooth mass average in Lemma 2.5. Choose the joint majority-and-L^p witness \mathcal B_R in Lemma 2.7, with that exceptional proportion. The bound d_R=o(1) is uniform in the stated range, so it lies below the corresponding fixed small-density threshold. Its moment bound is

This integral includes exceptional fibers and points with zero projection Jacobian.

Take the analogous witness \mathcal B_1 at scale s. For each of its entries, the matched projection entry on every good fiber is within Cs\sqrt\delta. The area formula and the uniform upper bound for J_\pi give these entries a fixed amount of G_s-weighted \mu-mass. Using |DV_b|\le C, |V_b|^2\le\Gamma_s, and the barrier U\le s^2\delta/k therefore gives

This estimate is made one entry at a time, so repeated entries require no disjoint allocation of mass. Match the centers of \mathcal B_R to those of \mathcal B_1 through concentric dyadic dilations. The majority sets have fixed positive relative overlaps, and the sum of the local matching bounds is at most C(1+R)^{K/2}s\sqrt\delta. For the fixed multiplier,

The last three bounds and the bounded normalized projected mass give

All these bounds use displacements from the fixed wells. Gaussian summation of (6.16) and of the energy dilation bound gives

The constants c_1 and C are fixed. To control more distant tails uniformly even when s is very small, note that clipping gives |v|\le C and the projected mass identity and dilation bound give (\pi_\#\mu)_{\sqrt2s}\le C. Gaussian comparison therefore gives

At R=C\sqrt a, with C sufficiently large, this is smaller than B^{-N}\delta^{p/2} for any prescribed fixed N. Likewise Gaussian comparison, with a slightly larger fixed dilation to absorb |u|^2, bounds the energy tail by Ce^{-cR^2}\delta. These estimates cover the part beyond the polynomial dilation range and the fixed chart cutoff; they introduce no negative power of s.

Take R_B=C_1\sqrt{\log B}, with C_1 fixed sufficiently large. Annular Gaussian summation using (6.16) makes both the flux tail outside B_{R_Bs}(x) and the tilt tail there smaller than o(B^{-\eps_0}\delta). Choose J to contain this ball in its interior, with \ell(J)\le C R_Bs. The remaining tilt has Gaussian integral at least cB^{-\eps_0}\delta. Since G_s\le Cs^{-m},

The upper density bound follows from the energy dilation estimate on a ball containing J. All resulting factors are fixed powers of \log B, and hence are at most B^{\eps_0} after increasing the threshold for B. This proves (6.13).

Let H be the union of the maximal dyadic subcubes of J with density larger than B^{C_*}\delta. They are disjoint, and the upper bound in (6.13) gives

It is essential here to estimate mass as well as volume. The identity \pi_\#\mu=Q\vol_S+M gives

In particular the mass defect has not been discarded. Writing \sigma=\tfrac12-\tfrac1p>0, the Gaussian-weighted mass of \pi^{-1}H is at most B^{c_2\eps_0-C_*}+B^{c_2\eps_0}\delta, after absorbing the volume ratio |J|/s^m into a power of B^{\eps_0}. Hölder’s inequality with exponents p,2,1/\sigma, together with |P_{vh}u|\le\sqrt e\,|u| and (6.17), bounds the absolute flux on this set by

Choose C_* large, independently of sufficiently small \eps_0, and then B large. Expression (6.20) is less than one quarter of the lower bound in (6.15). The strict inequality p>2 is used exactly at this step.

There remains flux at least cB^{-\eps_0}\delta over (J\setminus H)\cap B_{R_Bs}(x). Cauchy–Schwarz, the second-moment consequence of (6.17), and |u|^2\le R_B^2 on this core imply

Thus \nu(J\setminus H)\ge B^{-c_5\eps_0}\delta s^m. Compare with \nu(J)\le B^{\eps_0}\delta|J| and absorb the remaining fixed logarithmic factors. This proves (6.14) with a fixed c_*. The set J\setminus H is precisely the set in that formula, with the harmless choice of strict or non-strict threshold indicated there.□

Proof. We first prove the estimate for downward retracing of density. Let

For each j, stop at the first cubes, descending from the roots, whose density is at least \lambda_j. These high stopping cubes H are disjoint; a root is included if it already qualifies. Within each H stop at the first strict descendants R with d(R)\le e\lambda_j/R_*. These low stopping cubes are also disjoint, and

Now suppose P\supsetneqq P' are any two band cubes, meaning their densities belong to [\lambda_-,\lambda_+], with d(P')\le d(P)/R_*. Choose j so that \lambda_j\le d(P)<e\lambda_j. The cube P lies below a high stopping cube H, and P' lies inside a low stopping cube below that H. Hence the union E_{\mathrm{ret}} of all such low cubes has

This is a bound in \nu-mass. Moreover, if the ancestry of a terminal cube has such a retrace before reaching that cube, the entire terminal cube is in E_{\mathrm{ret}}.

We next construct a global forest of paying cubes. Its first generation consists of the first marked cubes below the roots. For each paying cube P, stop at the first strict descendants C with d(C)>Ld(P). By (6.21),

Below each such C take the first marked cubes, including C itself if marked, as the next paying generation. All cubes in one paying generation are disjoint. Induction gives

Consider a terminal F with a chain W_1,\ldots,W_N and with no band retrace in its ancestry before F. The first paying cube on its ancestry contains W_1. Suppose the jth paying cube P_j contains W_j. Both cubes are band cubes. Absence of a retrace implies d(W_j)>d(P_j)/R_* when they differ; the lower bound needed below also holds when they coincide. Consequently

The next density crossing, and then the next first marked cube, therefore occur at or before W_{j+1}. The (j+1)st paying cube contains W_{j+1}. It follows that F is wholly contained in a paying cube of generation N. Thus

Equations (6.24) and (6.25) prove (6.22). All families are countable; half-open cubes give the statement also for measures charging dyadic boundaries. Finite-depth truncation followed by monotone convergence is an alternative justification of every stopping construction above.

We apply Lemma 6.3 to all cubes which can occur in Lemma 6.2, marking a cube if it admits at least one of the witness labels. A label need not be selected coherently between different contact paths. Each such cube satisfies (6.21) with the uniform choices

Indeed Ld(J)\ge B^{C_*}e^{-a} for any of its labels. Take R_* =\exp(B^{1/4}) and S=\exp(B^{1/3}/2). The inequality S>LR_* holds for large B, and J_*=O(B).

There is a fixed number of grids. A fixed fraction of the ordered witnesses on each contact path belongs to one grid. Retain that subsequence and reverse its order, so that cubes are listed from outer to inner. By (6.4) and (6.1), consecutive radii differ by at least \exp(cB^{1/3}). The first two properties in (6.13) therefore give strict nesting, with the same contact point in every cube. The density bounds give the factor S for consecutive densities for large B. There are still N\ge cB^{1/2} cubes. Start the dyadic forest at its maximal cubes contained in the working chart; all witness cubes lie in this forest, and its roots are disjoint. Its total \nu-mass is at most d. Thus (6.22) is at most

Choose \eps_0 so small that c_*\eps_0<1/4. This proves (6.23) after increasing the threshold for B. The estimate concerns the union of full innermost cubes, not just the contact points or almost every point on selected infinite chains.□

Proof. Outside B_{L_0s_x}(x) the Gaussian kernels satisfy

Choose L_0 sufficiently large. By (6.26), at least c\theta/2 of the s_x-Gaussian integral is then inside this ball. Since G_{s_x}\le Cs_x^{-m}, the threshold balls B_x=B_{L_0s_x}(x) satisfy

Apply the usual decreasing-radius selection to these balls; see [19, Chapter 1, Lemma 3.4] for the blocking-ball property used below. For clarity, grouping radii in successive factors of two and taking maximal disjoint families gives disjoint selected balls B_i=B_{R_i}(x_i) such that every original ball meets a selected ball with radius at least half its own. Assign each contact to one such blocking selected ball; this gives both

The radius condition is part of the assignment, and does not follow merely from coverage of centers by enlarged balls. The radii are bounded, since c\theta\le\nu_{s_x}(x)\le C_m s_x^{-m}\nu(\R^m).

Within one assigned family apply the same selection to the derivative balls B_{u_x}(x). Obtain disjoint balls D_{ij}=B_{u_{ij}}(x_{ij}) whose fivefold enlargements cover its centers. All the original D_{ij} lie in B_{(3+2/L_0)R_i}(x_i), so comparison of ordinary volumes gives \sum_j u_{ij}^m\le C R_i^m. Positivity of the Gaussian on each fivefold ball and (6.26) give

Therefore the h outer mass of the contacts assigned to i is at most C\theta R_i^m\le C\nu(B_i), by (6.28). The B_i are disjoint and contained in E, which proves (6.27). This argument uses no absolute continuity of h, and remains valid if the derivative radii are much smaller than the threshold radii.□

Completion of the proof of Theorem 3.1. Fix Q and assume the assertion for all smaller positive integers. Choose C_Q large enough for Propositions 4.3 and 5.4 and Lemmas 5.2 and 6.1 and the bottom-contact estimate in Lemma 5.1. The lower-integer constants are already fixed. If Q=1, use only the one-well statements; no separated estimate, function b, or multiple-well construction is used.

Put F=F_Q(T), \eta=e^{-2F}, and a_0=F^2, as in Lemma 5.2. Suppose the localized source has mass at least \eta d. For each positive bottom scale use the entrance boundary K_\sigma, the supported contact set \mathcal S_\sigma, and the Borel threshold event \mathcal G_\sigma defined in Section 5. Write \widehat h=\widehat h_\sigma and h=h_\sigma=(\pi_X)_\#\widehat h; both have mass \int f. By Lemma 5.1, the \widehat h-mass of K_\sigma\setminus\mathcal G_\sigma is at most

as the bottom tends to zero. We estimate the contacts in \mathcal G_\sigma\cap\mathcal S_\sigma, whose \widehat h-mass equals \widehat h(\mathcal G_\sigma), uniformly in that bottom.

At such a contact let \underline s be its derivative scale and s_0\ge\underline s a threshold equality on the outward vertical. In particular this includes a contact on a vertical side below a grazing equality. The continuous stopping rule and localization give

The harmless cutoff errors can be absorbed in fixed constants in these relations. There is room for the fixed dilation \sqrt2s_0<\rho: for s\in[\rho/\sqrt2,\rho], the total mass bound gives \tau_s\le C\rho^{-m}d=o(e^{-a_0}), using a_0=F_Q(T)^2=o(T) and |\log\rho|=O(F_Q(T)+\log T). Thus (6.26) holds with \theta=e^{-a_0}, u_x=\underline s, and s_x=s_0.

For this periodized application, u_x\le s_x=s_0<\rho, and every fixed dilation of the balls used in the proof has a Euclidean lift inside the fixed coordinate chart for large T. The upper Gaussian comparison holds termwise over lifts outside the torus metric ball; at these radii G_s\le Cs^{-m}, and the lower bound on fivefold balls comes from the nearest lift. The two Euclidean selections and their volume comparisons therefore apply in that chart, with the radii already bounded by \rho.

Each such contact has a band and a grid supplying the ordered witnesses of Lemmas 6.1 and 6.2. For a band i and grid g, let A_{i,g}\subset\mathcal G_\sigma\cap\mathcal S_\sigma be the contacts admitting an eligible chain for that pair, and let E_{i,g} be the union of all its eligible full innermost cubes. The finitely many sets A_{i,g} cover \mathcal G_\sigma\cap\mathcal S_\sigma; they may overlap and need not be measurable. Each E_{i,g} is Borel because the dyadic family is countable, and Lemma 6.3 gives \nu(E_{i,g})\le C e^{-B_i^{1/8}}d.

For a contact in A_{i,g}, choose any one eligible chain. Its threshold ball lies in the innermost cube. Indeed a(s_0)=a_0+O(1), the smallest witness has level at least B_i\ge2a_0, and |a'|\le C. Hence

The cube contains B_{s_{\min}}(x), so it contains B_{L_0s_0}(x) for the fixed constant in Lemma 6.4.

For every subset A\subset K_\sigma, the induced outer measures obey

Indeed, a Borel spatial set V\supset\pi_X A has A\subset\pi_X^{-1}V and \widehat h(\pi_X^{-1}V)=h(V); take the infimum over such V. Apply Lemma 6.4 to the center set \pi_X A_{i,g}, choosing for each center any one contact in that class and its radii. Its threshold ball lies in E_{i,g}, and the Gaussian hypotheses were verified above for the full projection h. Thus repeated centers introduce no multiplicity factor, and h^*(\pi_X A_{i,g})\le C\nu(E_{i,g}). The covering proof uses no measurable choice of contacts or chains. Outer subadditivity now gives

The band choice in Lemma 5.5 ensures B_1^{1/10}\gg F^3. Consequently the right side of (6.30), and also Ce^{-a_0}d, are o(\eta d) as T\to\infty. First take T large with all fixed parameters chosen as above, and then take the bottom small in (6.29). These bounds contradict \widehat h(K_\sigma)=h(X)\ge\eta d.

Thus the localized source has mass less than \eta d. Restoring its subtractive margin, whose integral is at most C\eta d, proves

for sufficiently large T, uniformly over the stated nonnegative unit Lipschitz weights. This proves the induction step and hence Theorem 3.1 for every positive integer Q.□

Proof. We first give a local correction argument, keeping all its domains a fixed positive distance apart. Work in units of one fitting scale and normal coordinates over a smooth minimal input S'. Suppose that its slope and a sufficiently large finite number of its scaled derivatives are small, the mass is bounded, the selected integer is Q, and |z|\le Ca. Here and throughout this proof constants may incorporate fixed factors such as 2^{k_0}.

For the ambient normal displacement field Z(X)=X-\pi(X), direct differentiation gives

Testing stationarity with \chi^2Z, as in the classical height-to-tilt estimate [2, Lemma 31], absorbing the last term, and using |P_{vh}|^2\le e gives

Here H_{S'}=0 is essential: it removes the term linear in height. Lemma 2.5 then gives |M|\le Ca^2 on smaller balls. The integer is the previously selected Q: moving a normalized smooth mass test between the old and new projections changes its average by O(a), while the old mass defect is O(a^2). For small a its average therefore remains within 1/4 of Q. The exceptional projection counts occupy only O(a^2) of each such ball. Apply Lemma 2.5 successively through overlapping balls if the smaller region is not concentric.

Let u=Q^{-1}\pi_\#(z\mu), interpreted as a distributional normal section using \vol_{S'}. Normal stationarity and horizontal stationarity with a linear height multiplier imply

for compactly supported smooth normal fields v in a smaller region. Derivatives of u in this formula are distributional. For completeness, the trace of the normal test v\circ\pi is

Its terms of first order in (z,P_{vh}) are -\tr(A_zA_v)+P_{vh}:\nabla^\perp v; the constant trace is -H_{S'}\cdot v=0. The horizontal lift multiplied by z\cdot w identifies \nabla^\perp u with Q^{-1}\pi_\#(P_{vh}\mu) to first order. All remainders contain e, |z|\sqrt e, or |z|^2, against bounded derivatives of the test. Their mass is O(a^2). This proves (7.10), including the normal-connection terms. Its bilinear form can also be read directly from the quadratic part of the graph area integrand, whose differential blocks are (I-A_v,\nabla^\perp v).

Here is the local elliptic argument that turns (7.10) into an exact minimal correction. Choose an integer q>m/2+1 sufficiently large that the right side of (7.10) is bounded by the H^{q+2} norm of v, and choose a positive integer s>m/2+J+3. Fix a nested chain between the available radius 128 and the required radius 80, for example with intermediate radii 120,112,104,96,88, subdividing these intervals into finitely many further radii for the Sobolev bootstrap. Successive restrictions below use this fixed chain; all cutoffs and extensions have a positive buffer. For any nested pair of these working balls the Jacobi operator \mathcal J has a local right inverse \mathcal R with

Only finitely many coefficient derivatives are involved. One explicit construction removes boundary compatibility issues as follows. Embed the working balls in a fixed flat torus, extend the small coefficient perturbations of -\Delta by a fixed cutoff, and choose a smooth scalar bump b, supported outside the larger ball, with integral one. The operator -\Delta+b\int_{\mathbb T^m}(\cdot) is invertible: integrate the equation to obtain the mean of the unknown, solve the remaining mean-zero equation by Fourier series, and restore that mean. The Fourier multiplier gains two derivatives on every integer Sobolev scale. Its sufficiently small coefficient perturbation is invertible on each of the finitely many scales in (7.11) by a Neumann series. Multiply F by a cutoff equal to one on the desired smaller ball and apply this inverse. On that ball the added bump term is zero, so the asserted equation is exact.

Apply this construction to the residual \mathcal Ju. On a smaller ball, v_0=u-\mathcal R(\mathcal Ju) satisfies \mathcal Jv_0=0 and u-v_0=O(a^2) in H^{-q}. Moreover u=O(a) in that norm, since it is a measure of mass O(a). Interior regularity gives \|v_0\|_{H^{s+2}}\le Ca after another restriction. This last estimate follows directly from (7.11): apply the equation to nested cutoffs of v_0; each commutator has order at most one, so each step improves the available Sobolev order by one. Iteration reaches s+2 and then Sobolev embedding gives every fixed derivative bound required here.

Write the minimal graph equation for a normal section as \mathcal Jv+\mathcal N(v)=0. In H^{s+2} on these fixed balls, the product rule and s>m/2 give

Extend v_0 by a cutoff and solve w=-\mathcal R\mathcal N(v_0+w) on a ball of radius Ca^2 in H^{s+2}, putting the nonlinear cutoff equal to one on the ultimate working region. This is a contraction for small a. On that region v=v_0+w defines an exact minimal graph and v-u=O(a^2) in H^{-q}. Interior differentiation of the equation gives smoothness and any further fixed derivative estimates.

Converting this normal graph to a graph over P_0 gives the desired centered fit. Vertical displacement at the fixed horizontal base is a smooth linear transformation of z, up to O(|z|^2). Moving the argument of a smooth test from one projection to the other costs O(a^2) when multiplied by the height. Finally, evaluating the graph correction against \pi_\#\mu instead of Q\vol_{S'} costs O(a^3), by the mass-defect bound and its O(a) size. The first-order term is Q(u-v), proving (7.7). The same argument after changing to the new normal coordinates proves the normal version. The new graph has height error Ca and tilt Ca^2; another application of Lemma 2.5 gives its mass estimate.

We apply this procedure at j=0 using the horizontal plane as input, with initial errors small enough that (7.5) holds there. At j>0 use the preceding successful fit. Its test ball has radius 128\ell_j, whereas the new graph is required only on B_{80\ell_j}(x). Thus all the fixed intermediate balls just used fit inside the available region. Its height error in the new scaled units is at most 2^{k_0}m_0\ell_j^{k_0-1}. This proves the stated bounds at every constructed scale, including the first failure. The correction at scale \ell has physical derivatives of order i bounded by Cm_0\ell^{k_0-i}. Summing over the earlier dyadic scales is finite for i\le J<k_0. This maintains the small physical C^J bounds. The order choices here are noncircular: first fix J and the finite test order in (7.10), then q,s and the finite coefficient order required by (7.11), then k_0 and finally the smallness threshold for m_0. The required coefficient order may exceed J. Its uniform scaled derivative bounds follow by interior bootstrapping of each exact minimal input from its small slope and scaled C^{2,\alpha} bound, using the room between its domain and the new working region. The plane supplies this bound at initialization. Under the next halving of scale the second-derivative norm contracts by 1/2 and its Hölder seminorm by 2^{-1-\alpha}, while the correction adds only Ca_j and the slope increments Ca_j are summable. This closes the inductive bound before the higher-order bootstrap is applied. In particular the scaled C^{q+2} bounds used for changes of projection, multiplication of Sobolev tests, and the overlap argument below do not require physical derivatives through order q+2 to be included in J. Every threshold and constant is thus uniform along the construction.

It remains to prove the quadratic compatibility. Scale a common interior ball to unit size and put a=m_0\ell^{k_0-1}. For two fits f_1,f_2, the difference d=f_1-f_2 solves a homogeneous uniformly elliptic linear system: integrate the derivative of the minimal-graph flux along the segment between their gradients. Its coefficients have the bounded derivatives and small ellipticity perturbation already established. The height bounds on the support and the projection-count conclusion of Lemma 2.5 first give \|d\|_{L^1}\le Ca on an interior ball. Indeed, on a typical normal fiber of the first fit, choose a support point. Both fits are within Ca of that point at its horizontal coordinate. Moving back to the horizontal coordinate of the normal base point costs O(a) by the small slopes, so the difference there is Ca. The exceptional base volume is O(a^2), and the small slopes bound the difference there by a fixed constant once it is bounded at one point. Interior estimates for this homogeneous system therefore give \|d\|_{C^{q+2}}\le Ca on a smaller ball.

Subtract the two fixed-base moment estimates. This gives \int d\zeta\,\dd((\pi_0)_\#\mu)=O(a^2)\|\zeta\|_{H^q}, increasing q once if necessary. To compare this with base volume, first replace \pi_0X by \pi_0\pi_{S_1}X in the test d\zeta. The displacement is O(a) and d\zeta has derivatives O(a), so the error is O(a^2). Now Lemma 2.5 changes the measure to Q\vol_{S_1} with error O(a^3). Thus, writing J_{f_1} for the area density of the graph over P_0, J_{f_1}d=O(a^2) in a fixed negative Sobolev norm. Multiplication by J_{f_1}^{-1} preserves that bound. The same homogeneous interior estimates, now starting from this negative norm, give \|d\|_{C^J}\le Ca^2. Returning to physical coordinates gives (7.8).□

Proof. Suppose the assertion fails along a sequence m_0\downarrow0. Dilate each failed fitting ball by \ell^{-1} and put a=m_0\ell^{k_0-1}. Relative to its failed fit, |z|\le Ca and D+|M|\le Ca^2 on fixed interior balls. The reference graphs converge smoothly to a plane. Identify their bases by their graph coordinates, including their volume densities in the transported measures. After taking a subsequence on the connected interior disk B_{70},

Here \nu,d are positive, \beta is a positive semidefinite matrix-valued measure, and \mathfrak m is signed. The bound |z|/a\le C and \pi_\#\mu=Q\vol_S+O(a^2) in total variation imply \nu\le C\,\dd y on every compact interior set.

Failure means that the fixed-base vertical error exceeds a at a support point over B_{64}. Since the failed graph has uniformly small slope, its nearest normal distance is comparable to that vertical error. Thus |z|\ge ca at this point. The change of projection moves its base point by O(a), so a fixed interior ball about it still lies over B_{70}. By (7.9) and the vanishing curvature of the scaled fits, the tangential Laplacian of |z|^2 is bounded below by -o(a^2). More explicitly, absorb the e|A||z| term in the nonnegative e term and retain -C|A|^2|z|^2. Adding o(a^2)|X-X_1|^2 gives a nonnegative subharmonic function on a fixed interior ambient ball about that point. Its mean-value inequality from Lemma 2.1 shows that \nu(B_{70})\ge c>0.

The signed estimate gives the crucial additional inequality

We check its physical-scale hypothesis even if \ell stays bounded away from zero. Before applying Theorem 3.1, dilate the original physical space by \lambda_0=m_0^\sigma, where 0<\sigma<1/(J-1) and J is increased beforehand if further fixed coordinate derivatives are required. A physical derivative of order i\le J of a fit is now bounded by Cm_0\lambda_0^{1-i}, hence uniformly bounded. A comparable test ball has physical radius l\asymp\lambda_0\ell, and its physical height is at most C\lambda_0\ell a. With d_0=Ca^2 and T=-\log d_0, choose

Then l+\sup|z_{\rm phys}|\le d_0^\kappa for all sufficiently small m_0, uniformly in 0<\ell\le1. The fits are minimal, so their \Lambda consists only of physical height squared and \Lambda/a^2\le C\lambda_0^2\ell^2\pi_\#\mu\to0 locally. Apply Theorem 3.1 on finitely many interior balls to an arbitrary nonnegative smooth test, rescaling its size and Lipschitz constant by fixed factors. Divide by a^2 and pass to the limit. Its error is O(e^{-F_Q(T)}), which tends to zero, proving (7.13).

We next prove that \nu is nonzero in every open interior subball. This is a finite-radius unique-continuation argument for the limiting measures, and requires no regularity theorem for a multiple-valued limiting function. Fix an interior center y_0 and an integer p\ge4, and set

In this paragraph only, define

Dots denote differentiation in \log s. Stationarity gives

Here are the identities before the limit. The horizontal test (y-y_0)\phi_s gives the first equation. The normal test z\phi_s identifies s^2\int\phi_s e/a^2\,\dd\mu with

The horizontal test (y-y_0)\psi_s|z|^2/a^2 gives twice this flux for the derivative of the height integral. Terms containing both tilt and |z|^2/a^2 have mass O(a^2), and the geometric errors tend to zero by (7.9) and Lemma 2.2. Finally P_{hv}P_{vh}=B-B^2\le B and Cauchy–Schwarz bound the square of the flux by the product defining \mathcal R\mathcal H. The last inequality is (7.13).

Suppose there is a positive first radius s_0 at which this height becomes nonzero. Put t=\log s, t_0=\log s_0, and choose a fixed t_1>t_0 with all the balls strictly interior. On (t_0,t_1] put P=\mathcal A/\mathcal H and n=\mathcal D/\mathcal H. Then

For P\ge1 the last expression is nonnegative on n\ge2P. Consequently P(t)\le\max(1,P(t_1)) going inward. On the zero-height core, the second identity in (7.15), applied also to smaller balls and other centers, gives d=0. To pass the total-variation estimate of Lemma 2.5 to this core, reuse the auxiliary physical dilation \lambda_0=m_0^\sigma above: it leaves normalized M,D unchanged and makes a^{-2}\Lambda\to0. For a ball B with 3B compactly inside the core, take \chi\in C_c(3B) equal to one on 2B. For every \zeta\in C_c(B) with |\zeta|\le1, that estimate gives

Hence \mathfrak m=0 in the core, without a boundary-continuity assumption on the limiting measures. The compact kernels converge uniformly as t\downarrow t_0 and vanish on the boundary of the core, so \mathcal A(t)\to0 even if \mathfrak m charges that boundary. Moreover

because n^2-n\ge-1/4. The function \mathcal H is nondecreasing. Integrating from t_0 therefore gives the explicit lower bound

With P bounded on this finite interval, (7.16) implies \dot P\ge n^2/2-C. Integration gives a uniform bound for \int_{t_0+\epsilon}^{t_1}n^2\,\dd t. Cauchy–Schwarz and (\log\mathcal H)^{\displaystyle\cdot}=2n now preclude \mathcal H(t)\to0 at t_0, a contradiction. If \nu vanished in an open ball but were nonzero elsewhere in the connected interior disk, choose a point of the zero set sufficiently close to its boundary. Its distance to \spt\nu is positive and smaller than the available interior radius, producing precisely the excluded s_0.

We also need strong, rather than merely distributional, information about the first moment. Divide the first moment by a and write U_\alpha=\pi_\#((z_\alpha/a)\mu). The horizontal height-multiplier test has, in the limiting flat coordinates, the distributional form

In the flat case one may take F_{i\alpha}=\pi_\#(P_{v_\alpha h_i}\mu/a) and E_{ij\alpha}=\pi_\#((z_\alpha/a)B_{ij}\mu). Cauchy–Schwarz bounds the first mass by C and the second by C\int e\,\dd\mu=O(a^2). In the actual coordinates, multiply the identity by the smooth inverse base metric and volume density. The additional terms have the same form: coefficients converge smoothly, flux masses remain bounded, and the terms containing height times tilt have mass tending to zero.

Take the divergence in (7.17) and solve on a smaller ball by a cutoff Newtonian potential. First derivatives of the Newtonian kernel convolved with bounded measures form a precompact family in local L^1: their local kernels are integrable and have a uniform L^1 translation modulus. The second-derivative potential of E tends to zero in measure by the weak (1,1) estimate proved in Lemma 2.4. The remaining harmonic distributions are compact on interior balls by their bounded negative norms. Thus U is precompact locally in measure after replacing it by its projection-count density. This replacement is legitimate: the area formula and the Jacobian estimate give

One can make the potential argument for functions throughout by mollifying with a radius chosen separately for each member of the sequence so that the count density changes by o(1) in L^1. The above small total-variation difference remains o(1) after mollification. The densities are uniformly integrable, since |z|/a\le C and their total counts differ from Q by o(1) in L^1. Finally (7.7) says that their distributional limit is zero. Precompactness in measure and uniform integrability therefore show that the normalized count first moment tends to zero in measure (indeed in local L^1).

On each fixed subball, the limiting second moment is positive. Fibers of total count different from Q have vanishing count mass by Lemma 2.5; points with tilt exceeding a number tending slowly to zero have vanishing mass by (7.6). Discard both. On a remaining fiber list its Q normalized heights with repetitions as w_1,\ldots,w_Q, and set \bar w=Q^{-1}\sum_iw_i. The identity

and |w_i|\le C show that a positive amount of the second moment is variance. Hence a subset of fixed positive base volume has variance bounded below, and therefore has two heights separated by a fixed positive amount. Their distinct spatial entries each have multiplicity less than Q. Restoring lengths gives the asserted separation. For Q=1 the variance is identically zero, so failure is impossible. If uniform constants for the prescribed comparable subballs failed, their centers and radii would have a convergent subsequence in the compact permitted range; the preceding argument on an interior subball of its limit gives the same contradiction.□

Proof. Let \ell_x be the first failed scale at x\in B_5, and put \ell_x=0 if there is no failure. Define

Every function in this supremum is 1-Lipschitz, so t is 1-Lipschitz and \mathcal I is the union of the failure balls.

We describe the gluing before proving the density properties. For \epsilon>0 use t_\epsilon=\max(t,\epsilon). Choose a covering by balls of radius c t_\epsilon(y_i), with c>0 a sufficiently small fixed constant, by first taking disjoint balls of a smaller such radius and then enlarging them. The Lipschitz bound makes radii of intersecting enlarged balls comparable; the covering has bounded overlap. Smooth bumps on these balls, divided by their sum, give a partition \{\chi_i\} with |D^j\chi_i|\le C_jt_\epsilon(y_i)^{-j}. At y_i choose a dyadic constructed scale L_i between C_0t_\epsilon(y_i) and 2C_0t_\epsilon(y_i), capped at the initial scale 1. The constant C_0 is fixed and the covering radii can be reduced to leave all required overlap room. Since \ell_{y_i}\le t(y_i), these scales have been constructed. Where the cap is active, L_i\asymp t_\epsilon(y_i) still holds, with a constant depending on C_0. Let f_\epsilon=\sum_i\chi_i f_{y_i,L_i}.

On an overlapping cell of size L, fix one of its fits f_* and write f_\epsilon-f_*=\sum_i\chi_i(f_i-f_*). By (7.8), every derivative of order j\le J+1 in this identity is at most Cm_0^2L^{2k_0-1-j}, by the choice of derivative order J+1 in Lemma 7.1. Each f_i is minimal. The mean-curvature operator and its first derivative involve at most three derivatives of the graph. Substitution therefore gives

The support height is at most Cm_0t_\epsilon^{k_0}, by the height bounds for the fits. Uniform C^{J+1} bounds give, along a sequence \epsilon\downarrow0, a local C^J limit f. Passing to the limit proves (7.18), including at t=0. Uniform small slope supplies a normal tube on each smaller chart. The change in horizontal argument under nearest projection is at most Cm_0|z|\le Cm_0^2t^{k_0}, which is a small fraction of t when t>0. The Lipschitz bound on t therefore proves the assertion about the projected argument, including the zero set by a limit.

We next use multiplicity defects to control the tents. Choose the comparable subball in Lemma 7.2 strictly inside the original failure ball. Small slope and the much smaller height displacement then imply, in the fixed horizontal projection,

This charges inside the original ball, a fact needed in the disjoint covering below. The change from normal to horizontal projection has a uniformly bounded Jacobian and displaces the chosen smaller subball by much less than its interior margin.

Fix a large finite K, greater than every fixed enlargement factor that will occur in the following local arguments. At initialization choose \eta<c/(K+1)^m in (7.4). If |x|\le K\ell, then

a contradiction. The enclosing ball is inside the fixed chart; one uses its actual radius |x|+\ell, not an unnecessarily larger ball. Thus |x|>K\ell for all relevant failures, and t(0)=0. If B_{A\ell}(x) meets B_r(0), for a fixed A<K, then

Combining (7.21) with the little-o part of (7.4) now gives

All limits here hold for the one fixed choice of the initial data. This first excludes large intrusions using a uniform bound and only then uses differentiation to obtain the little-o conclusion.

For small r, all original failure balls meeting B_r lie in B_{2r} by (7.22). Select disjoint original balls B_i whose fivefold enlargements cover their union. The charges in (7.21) are disjoint, so

The same small-failure estimate gives \sup_{B_r}t=o(r). The support contains 0, so its zero height at t(0)=0 shows that S passes through 0. The superlinear height bound and (7.1) force T_0S=P_0. Smooth change of coordinates transfers the volume statement to S, and the projected height bound gives |z|\le Cm_0o(\rho)^{k_0}\le C\rho^2.□

Proof. We give the geometric estimate that pays for points close to the boundary \rho=r. If t(y)>0, choose a failure (x,\ell) for which

Then \ell>t(y)/2, |x-y|<\ell, and the Lipschitz bound gives t(v)\le(2+A)\ell when |v-y|\le A\ell. An upper comparison of \ell with t(y) is neither asserted nor needed. On this whole neighborhood the support has height at most Cm_0\ell^{k_0} over S, and its mass over any ball of radius comparable to \ell is at most C\ell^m by the ambient upper mass bound.

In each fixed small comparable subball within the permitted distance from x, Lemma 7.2 gives mass at least c\ell^m with distance from S at least cm_0\ell^{k_0} and with tilt relative to S less than, say, 1/4. To verify this transfer, the normal fiber of the failed fit is transverse to the small-slope graph S. It has only one intersection with that graph in the working tube, and distance to S is comparable to distance along the fiber to that intersection. Of two separated entries at least one is a fixed fraction of their separation from S. Their tilt relative to S is still small, since both reference graphs have small slope. Restrict the subball slightly first: all changes of projection argument are o(\ell) and so preserve its interior margin. On this mass h_r\ge3/4 after reducing the tilt threshold in Lemma 7.2 once.

For y with \rho(y)<r, the witnessing scale satisfies \ell=o(r) by (7.22). There is a ball of radius c_1\ell at distance O(\ell) from y, lying inside \rho<r, on which

Here C is a cell of side c_2\ell containing y, and the suprema concern its intersection with \rho<r. If y is already a fixed multiple of \ell inside the boundary, take a nearby subball there. Otherwise move its center inward along the radial geodesic by C_1\ell and take c_1 much smaller than C_1. The boundary distance on the new ball is at least a fixed multiple of the largest boundary distance on the cell and at least c\ell. Since 1-\rho^2/r^2 is comparable to (r-\rho)/r near the boundary, this gives (7.29); in the central half ball all weights are bounded below. The displacement is bounded by a fixed small multiple of \ell, so the new ball is among those allowed in Lemma 7.2. These assertions are unchanged by the smooth base-coordinate conversions. Thus they also apply to cells straddling \rho=r.

The separated mass on this inward ball gives

If the support fails to lie on S in every neighborhood, then \delta(r)>0 for every small r. Choose a point with height at least half the defining supremum. Its height is nonzero, so it has a witness as in (7.28); (7.18) and (7.30) give

The inward ball has mass at least c\ell^m, height at least cm_0\ell^{k_0}, and h_r\ge3/4. Consequently

In the last step use \ell/r\ge c(m_0)\delta(r)^{1/k_0}r^{1/k_0-1} and r\le1. The upper bound in (7.25) follows from bounded normalized mass and h_r\le1. The bound \delta(r)\le Cr follows from Proposition 7.3. This proves (7.25).

For any point with t>0, its witness and (7.30) give

At t=0 the left side is zero. This proves (7.26).

Finally group points with t>0 according to the dyadic scale of a witnessing failure. At scale \ell=2^{-j} cover the group by grid cells of side c_2\ell, taking each grid cell only once. For each occupied cell choose a witness from that group and an inward ball as above. On the cell the curvature-square integrand is at most Cm_0^4\ell^{4k_0-8}. Its mass is at most C\ell^m. On the corresponding inward ball the integral of \phi_r|z|^2 is at least c m_0^2\ell^{2k_0+m}\sup_C\phi_r by (7.29). Hence

The displacement from a cell to its inward ball is O(\ell), so these balls have bounded overlap at each fixed dyadic scale. Sum first over the cells and then over j. The geometric series \sum_j2^{-j(2k_0-8)} converges because k_0>4. Points with t=0 contribute nothing, and division by r^m gives (7.27).□

Proof. Write the restricted varifold as \vartheta\,\HH^m\llcorner S, allowing the integer density \vartheta to be zero off its original support. This is the standard constancy argument; compare [19, Chapter 8, Theorem 4.1]. Tangential first variation gives \int_S\vartheta\,\Div_S X=0 for all compactly supported tangent fields. Thus \vartheta is constant almost everywhere on a connected smaller disk. The constant is Q, by (7.1). Normal first variation gives H_S=0, and the small-gradient minimal graph equation bootstraps S to a smooth graph. The origin is therefore regular.□

Proof. For the absolute estimate it suffices to treat \|a\|_{1,r}\leq1 uniformly: apply that case to a/\|a\|_{1,r} when the norm is positive, and the zero case is immediate. The signed estimate will use the member a=1 of this unit class. Choose a small fixed \beta>0, with \beta<1/4, and put \sigma=\delta^\beta; we may take 0<\alpha\leq2\beta in \eta_1. Choose a radial cutoff \chi equal to one when 0\leq1-\rho/r\leq\sigma^2, equal to zero when 1-\rho/r\geq2\sigma^2, and with derivative bounded by C/(\sigma^2r). Write \phi_{\rm thin}=\chi\phi and \phi_{\rm bulk}=(1-\chi)\phi.

We first treat the bulk, including the overlap with the thin part. Fix first an enlargement factor c_{\mathrm{out}}>8 sufficient for Lemma 2.5 and Theorem 3.1 and for a Caccioppoli cutoff equal to one on B_{8\ell} and supported in B_{c_{\mathrm{out}}\ell}. Then choose the fixed c_0 small and put \ell=c_0\sigma^2r. Cover the bulk by balls of radius \ell with centers x_i in its support. These centers have boundary depth at least \sigma^2r, so c_0 can be chosen so that all B_{c_{\mathrm{out}}\ell}(x_i) stay inside B_r. Choose the cover with bounded overlap of these enlargements and so that the intermediate balls on which Lemma 2.5 selects integers have overlaps of fixed positive relative volume along the connecting chains. Choose a smooth nonnegative partition of unity \{\vartheta_i\} subordinate to the original balls, with

where D denotes base-coordinate derivatives and k=k(m) is the finite test order in Lemma 2.4. Put \phi_i=\phi(x_i). On each enlargement \phi is comparable to \phi_i. The partitioned weights, divided by \phi_i and by a fixed constant, have Lipschitz norm at most one in units of \ell.

The mass on each enlarged ball is at most C\ell^m. To see this without assuming a local graphical representation, observe that \delta r=o(\ell); thus the part of the tube over the ball lies in an ambient ball of radius C\ell centered at a support point. The upper bound follows by monotonicity, comparing with an ambient ball of radius comparable to r and using the fixed normalized mass bound at the origin. Empty balls need no estimate.

For the local tilt estimate, use ordinary, unnormalized physical integrals. Let \xi be a base cutoff equal to one on B_{4\ell}, supported in B_{c_{\mathrm{out}}\ell}, and satisfying |\nabla_S\xi|\leq C/\ell. The analogous cutoff equal to one on B_{8\ell} has the same derivative bound and outer support. Testing stationarity with \xi^2z and using the normal-displacement trace (7.9) gives

Indeed, the derivative of the cutoff contributes at most C\xi|\nabla_S\xi||z|\sqrt e; Young’s inequality absorbs this term and the small |\mathrm{II}||z|e term from the trace. The displayed |H_S||z| term is retained because the center need not be minimal. The bounds |z|\leq\delta r and |H_S|+|\nabla H_S|\leq C\delta r, together with the local mass bound, give \ell^{-m}D(B_{4\ell})\leq C[(\delta r/\ell)^2+(\delta r)^2]. The definition of \Lambda gives \ell^{-m}\Lambda(B_{4\ell})\leq C(\delta r)^2 separately. Hence

The same estimate holds on B_{8\ell}, using its cutoff in B_{c_{\mathrm{out}}\ell}. These enlargement factors were fixed before c_0. The coordinates have the required fixed geometry bounds, and the terms in \Lambda still use physical units.

Apply Lemma 2.5 first to select the local integers. On the intermediate balls just chosen, the scalar exceptional sets have relative volume O(\delta^{2-4\beta}). For small r they cannot fill the chosen overlaps of fixed positive relative volume, so their integers agree. The bulk is connected; call the common integer q_r. To identify it, rescale one fixed smooth nonnegative test supported in B_{1/2} to B_{r/2} and normalize its reference-volume integral to one. Call it \varphi_r; then |D^j\varphi_r|\leq Cr^{-m-j} for 0\leq j\leq k. Partition this test over the cover. After pullback to a unit \ell-cell, each product \vartheta_i\varphi_r has C^k norm at most Cr^{-m}, since \ell\leq r. The coordinate-volume factor in the scalar proof has fixed scaled derivative bounds through order k, so multiplying by it preserves this estimate. The distributional estimate (2.14) used in the proof of Lemma 2.5, with (8.6), bounds each cell’s error relative to q_r by C\delta^{2-4\beta}\ell^m r^{-m}. Bounded overlap gives \sum_i\ell^m\leq Cr^m, so its projected-mass average differs from q_r by at most C\delta^{2-4\beta}. Tangent convergence and the smooth rescaling of the fixed center make that same average Q+o(1). Integrality therefore gives q_r=Q for all sufficiently small r. Exact agreement on overlaps involves no error accumulated along the chain of balls. This also identifies the integer on the balls used in the overlap and in the thicker error collar below. Consequently Lemma 2.5 bounds |M| on each smaller ball by the enlarged D+\Lambda mass. In particular, applying it with fixed interior room and using (8.6) also on the larger ball gives

All these balls lie inside B_r by the choice of c_0. Summation with comparable weights gives

using (8.4) to absorb a small multiple of E. For every member of the unit multiplier class this also gives r^2|\int^{(r)}a\phi_{\rm bulk}\,\dd M|\leq C(E+r^2H).

For the signed estimate, invoke Theorem 3.1 afresh on each normal chart at physical radius \ell. The measures M,D,\Lambda and the base tests are geometric, so they give the required data in the unit coordinates, with the measures divided by \ell^m. The physical condition below belongs to this new theorem instance; it does not change the inherited geometry parameter in an off-center rescaling of a prior instance. On a cell use

with the fixed enlargement and constant sufficient for Theorem 3.1. Equation (8.7) shows that c|\log\delta|\leq T_i\leq(2J_0+4)|\log\delta|, where the upper bound follows from the floor in d_i. Moreover

for some fixed \kappa>0 and all small r; choose, for example, \kappa(2J_0+4)<2\beta. Thus both smallness hypotheses of Theorem 3.1 hold even if \delta is smaller than every fixed power of r.

Apply that result to w_i=\vartheta_i\phi_{\rm bulk}/(C\phi_i) in the coordinates scaled by \ell. They are supported in B_1 in those coordinates and satisfy 0\leq w_i\leq1 and \Lip w_i\leq1. Restoring the physical measure factor \ell^m and then the normalization r^{2-m} gives

The last sum includes the enlarged cells, even where \phi_{\rm bulk} vanishes in the overlap with the thin part. On these cells use the local estimate (8.7): \phi is comparable to \phi_i throughout B_{8\ell}(x_i), and these balls have bounded overlap inside B_r. Therefore

Here the last floor sum is bounded by the ordinary volume of the bounded-overlap cover. Since T_i\geq c|\log\delta| and \exp[-F_Q(c|\log\delta|)] is smaller than any fixed negative power of T, the remainder in (8.9) is at most

The floor term r^2\delta^{2J_0+4} is at most C\delta^{J_0+4}H by (8.2). The weighted comparison (8.4) controls the remaining integral by C(E+r^2H) and bounds the C_Q\Lambda term in (8.9) by Cr^2H+o(g)E, since r^2\delta^2=o(g). This proves the second line of (8.5) for the bulk, with its leading tilt term unchanged.

It remains to estimate the signed mass on the thin part. A direct total-variation estimate here would unnecessarily add a full-tilt term. Instead solve a radial divergence equation. In geodesic polar coordinates write the volume element as j(\rho,\omega)\,\dd\rho\, \dd\omega. Throughout this thin calculation \|a\|_{1,r}\leq1, and all constants are uniform over that class. Initially put

Then \Div_S X=a\phi_{\rm thin}. Multiply X by a radial cutoff equal to one at depths r-\rho\leq\sigma r and zero at depths r-\rho\geq2\sigma r. On this latter collar the integrated numerator has size O(r\sigma^{2p+2}) relative to j. The cutoff divergence error q_c consequently satisfies

and is supported where \sigma r\leq r-\rho\leq2\sigma r. The same integration, also differentiated in angular directions, gives

For example, at depth d r\leq2\sigma^2r the primitive divided by r is O(d^{p+1}), whereas \phi\asymp d^p; beyond that depth its size is O(\sigma^{2p+2}). On the cutoff collar the derivative loss \sigma^{-1} still leaves \sigma^{p+1}\phi. This proves (8.10); bounded derivatives of the polar volume factor only change its constants. A radial spatial derivative of the primitive uses the value of a at its lower endpoint; an angular derivative uses only its first spatial derivative, bounded by 1/r. The scale parameter r is held fixed in this calculation. Near \rho=r, the preceding estimates give |F|\leq Crd^{p+1} and |\nabla_S(F\mathbf u)|\leq Cd^p. Thus X and its first derivative vanish at the boundary, so extending X by zero gives a C^1 compactly supported base field using only values of a in B_r. The inner cutoff removes the polar origin. When m=1 the construction is made separately on the two radial intervals and has no angular term.

Use the lifted field of Lemma 2.2. The q_cM term is estimated by the already proved bulk total-variation bound: for small \sigma, its support has depth between \sigma r and 2\sigma r, where \chi=0 and \phi_{\rm bulk}=\phi. Its contribution is at most C\sigma^{p+1}(E+r^2H). Put W=\int^{(r)}\phi|z|^2\,\dd\mu\leq C(H+\delta^2E). Equation (8.10) gives |X|\leq C\sigma^2r\phi and |\nabla_SX|\leq C\phi. After multiplication by r^2, the three error types in Lemma 2.2 are therefore bounded by

using the weighted curvature-square bound for the term containing \nabla^\perp H_S. Young’s inequality puts these terms into Cr^2H+C(\delta r+\sigma^2)E. The stress left after its radial part costs C\sigma^2E. Since p\geq4, r\leq1, and 0<\alpha\leq2\beta, all these small coefficients of E are at most \eta_1 after fixing C_1. The main stress term gives

Since \rho\geq r/2 on the thin part,

Thus the first term of (8.11) is at most C\sigma^2R+C\delta^2r^2E\leq\eta_1(R+E), after increasing C_1. Adding the unit bulk bound gives

Homogeneity now gives the first line of (8.5) on the same radius interval. For the signed estimate use a=1, add twice its thin-part bound to the bulk signed estimate, and use \int\phi_{\rm bulk}\,\dd D\leq\int\phi\,\dd D. Because \eta_1=o(g) by (8.2), its E term is absorbed into CgE after reducing the fixed radius. Enlarge C_1 once to cover the fixed factors in both conclusions. The coefficient one has been preserved as claimed.□

Proof. Test normal stationarity with z\phi. The exact normal-displacement trace formula (7.9) and (8.4) give

In particular the H_S\cdot z term is bounded using the weighted curvature-square estimate, rather than a height-independent error. For the tangent projection P_T, the algebra is exact: P_T^2=P_T and (P_T)_{hh}=I-B imply

Cauchy–Schwarz therefore bounds the absolute value of the flux in (8.15) by \sqrt{RH}. Division by H proves (8.12).

For the horizontal identity put Z_0=\rho\mathbf u on S and let \overline Z_0(q)=J_zZ_0(\pi q) be its unweighted lift. We first compute the full derivative of the kernel in the test (\phi\circ\pi)\overline Z_0. Since D\pi=K_zP_h and \nabla_S\phi=-(\rho/r^2)\psi\mathbf u,

Both this vector and \overline Z_0=\rho J_z\mathbf u are horizontal. The product rule and (P_T)_{hh}=I-B therefore give exactly

The identity coefficient is one because J_z,K_z are self-adjoint inverses and (K_z\mathbf u)\cdot(J_z\mathbf u)=|\mathbf u|^2=1. The radial terms are understood to be zero on the central fiber, as their factors of \rho^2 require.

For the remaining derivative, the pointwise block calculation in the proof of Lemma 2.2 gives

This is the estimate for the unweighted lift, so the product formula multiplies its entire remainder by \phi. Subtract the reference identity Q\int_S\Div_S(\phi Z_0)\,\dd\vol_S=0, in which \Div_S(\phi Z_0)=\phi\Div_SZ_0-\rho^2\psi/r^2. Since \dot A=(2-m)A+\int^{(r)}\rho^2\psi\,\dd M, we obtain, with

the exact identity

In geodesic normal coordinates y, Gauss’ lemma gives Z_0=y^i\partial_i. If g^S_{ij} are the metric components, then

The logarithmic volume factor has vanishing first derivative at 0 and bounded second derivative in the fixed chart. Consequently

for one fixed K_{\mathrm{geo}}. Also \nabla_SZ_0=I+O(\rho^2). The number K_{\mathrm{app}}=\max\{1,K_{\mathrm{geo}}\} bounds both actual collar multipliers: a_r here and a=1 in the bootstrap. Choose C_\eta in the convention above to dominate K_{\mathrm{app}}C_1 and the finitely many other error prefactors in the frequency estimates. Then \eta\geq K_{\mathrm{app}}\eta_1 has a fixed prefactor, so (8.3) applies. The first line of (8.5) and positivity of B now give

For the pointwise remainder use |Z_0|\leq r, |\nabla_SZ_0|\leq C, and \sup|z|\leq\delta r. The first and last lines of (8.4), followed by Cauchy–Schwarz, bound its three terms respectively by

Here the term involving \nabla^\perp H_S uses the weighted curvature-square bound inside \Lambda; the last term comes from \sqrt e\,|z|\,|Z_0|.

It remains to compare the radial stress \widetilde R with R. Positivity of B, its trace e, and |J_z-K_z|\leq C|z| imply |B[v,(J_z-K_z)\mathbf u]|\leq C|z|\sqrt{be}. Thus |\widetilde R-R| is at most

by Cauchy–Schwarz. Since e\leq m, the second line of (8.4) gives

Combining these estimates in (8.16) and dividing by H proves (8.13). The differentiated kernel has been exhausted by \widetilde R; every other remainder in that identity carries the weight \phi.

Finally, set

This field is defined independently of the varifold tangent plane. Since \nabla\rho=K_z\mathbf u, it satisfies Y\cdot\nabla_T\rho=\rho h_r. Testing stationarity with \psi|z|^2Y consequently cancels the differentiated kernel exactly and gives

There is no derivative of the measurable tangent projection in this calculation. Also \nabla|z|^2=2z in the ambient Euclidean space, so normal-frame connection terms do not arise from this factor.

The center estimates give t(y)=o(\rho(y)) and hence |z|\leq C\rho^2 on the support near the origin. Although derivatives of \mathbf u have size O(\rho^{-1}), differentiation of Y thus gives, on the support,

It follows that \tr_TDY-mh_r=m(1-h_r)-e+O(r). The positive m(1-h_r) contribution is bounded by C\delta^2(E+R), by the proof of (8.4). The O(r) contribution is at most CrH+\eta(E+R), and the term -e|z|^2 can be discarded for an upper bound. Replacing the denominator in the second term of (8.18) by one costs at most C\delta r\sqrt{RH}\leq CrH+\eta R. Equation (8.15) now proves the upper bound in (8.14).

To justify the test at the central fiber, first remove \rho<\epsilon by a smooth radial cutoff \chi_\epsilon, equal to zero for \rho\leq\epsilon, equal to one for \rho\geq2\epsilon, and satisfying |\nabla\chi_\epsilon|\leq C/\epsilon. Since |Y|\leq C\rho, |Y||\nabla\chi_\epsilon|\leq C on the transition annulus. Its extra term is therefore bounded, at fixed r, by a constant times r^{-m}\int_{\rho<2\epsilon}|z|^2\,\dd\mu, which tends to zero. The preceding support bound and local mass bounds also justify all other limiting terms. The lower bound in (8.14) follows directly from differentiating r^{-m}\psi_r: its derivative in \log r is at least -m r^{-m}\psi_r, and the remaining factor |z|^2h_r is nonnegative and independent of r.□

Proof. All quantities are locally absolutely continuous as functions of \log r where differentiated; H>0, and inequalities for derivatives may be read almost everywhere. Put v_0=\dot A/H and h_0=\dot H/H temporarily. The first line of (8.5) with a=1, followed by (8.12), gives

It follows, for a sufficiently large fixed K and |P|\geq K, that

For the first assertion, either C\sqrt{q_*} or C\eta q_* is at least a fixed fraction of |P|. The second follows from (8.13), increasing K and then shrinking the radius so that q_* absorbs 2|P|, n_*, and every error.

If P\leq-K, the lower bound h_0\geq-m gives

Such an excursion persists as the radius decreases: in inward time s=-\log r, P_s<0 throughout this region. There also A<0 and \dot A>0, so A decreases inward and stays below a fixed negative number. This contradicts A\to0, which follows from its factor r^2 and bounded normalized mass. We conclude that P\geq-K on the small-radius interval.

There are arbitrarily small radii at which P\leq T^2. Otherwise, eventually P>T^2, and (8.20) and (8.3) imply v_0/P\geq cT^2. Define

Inward differentiation gives Z_s=v_0/P-2\geq cT^2-2. Since T\geq2s-C, Z becomes positive and unbounded. Also Z\leq T in this regime, whence eventually Z_s\geq c'Z^2. Integrating (1/Z)_s\leq-c' forces a blow-up at finite inward time, although A,H are defined and positive throughout every finite subinterval of this regime. This is impossible.

We next prove the differential estimate

Combining (8.13) and (8.14) gives

If q_*\geq L P^2 for a sufficiently large fixed L, then n_*\leq C\sqrt{q_*}+Cr^2 and \eta T^4=o(1) show that the right-hand side is positive. Otherwise n_*\leq CP, and squaring (8.12) gives

The independent signed second line of (8.5), with \eta_1\leq\eta, gives

For large K and small r, this also gives n_*>1. The principal terms factor as

Thus \dot P\geq-C(rP+gP^2+\eta P^3). The last term is absorbed by gP^2, since P\leq T^4 and \eta T^4\leq g after shrinking the radius. This proves (8.21).

It remains to justify that this restricted estimate gives a bound on every smaller radius. Choose an arbitrarily small moderate radius r_0=e^{-s_0}, so P_0\leq T_0^2. Take s_0 so large that

This is possible by (8.3). On each component where P\geq K, before a possible exit through P=T^4, (8.21) yields

Moreover Z_s>0 there by (8.20), increasing K if necessary. If the first component starts at s_0, these inequalities give

For large T_0 these bounds exclude P=T^4. If a component starts later, at s_i, then P(s_i)=K, and the same argument gives P\leq e^{\varepsilon_0}K and T\geq T(s_i) there, again excluding the upper exit. This proves the finite uniform bound

Write this bound as |P|\leq B_0. After B_0 is fixed, choose 0<\bar r_0\leq r_0 so that C\eta(1+B_0)\leq1/4 for the constants in Lemma 8.2 whenever 0<r\leq\bar r_0; this absorbs its \eta q_* terms after quotient differentiation. For P\geq0 use the upper height derivative bound in \dot P=v_0-Ph_0; for P<0 use its lower bound. In both cases Lemma 8.2 and n_*\leq C\sqrt{q_*}+Cr^2 imply

Integrating over a logarithmic interval of fixed length bounds \int q_*\,\dd\log r. Cauchy–Schwarz then bounds \int n_*\,\dd\log r, and integration of (8.14) proves both sides of the height comparison in (8.19). In each interval [2^{-j-1}\bar r_0,2^{-j}\bar r_0] choose a radius where q_* is bounded by its logarithmic average. Equation (8.12) bounds n_* at those radii, proving the last assertion.□

Proof. Take the radii r_j from Proposition 8.3, and write r=r_j. On \rho<3r/4 the weights \phi,\psi have positive lower bounds depending only on p. Equations (8.4), (8.12), and the boundedness of n_*,q_* give

Here a height integral is over the part of the separated tube above the indicated base ball, whereas D,\Lambda are projected measures. On the other hand, applying (8.19) with \lambda=1/4 and using \psi\leq2p, h_r\leq1 at radius r/4 gives

Fix an exceptional fraction \delta_{\mathrm{cfg}}=1/4 and a finite exponent p_*>2 satisfying m(1/2-1/p_*)<1. Use the enlargement c fixed from the local data in Lemma 2.7, with the required scalar-selection room, and then fix d_{\mathrm{cfg},0}=d_0(\delta_{\mathrm{cfg}})>0. Then choose \epsilon>0 so small that B_{r/4} has a finite cover by balls of radius \epsilon r whose required enlargements lie in B_{r/2}. The number of balls and \epsilon are fixed independently of r.

Apply Lemma 2.5 with room between 3r/4 and r/2; its integer is again Q. Its variation estimate and (8.23) give, for every enlarged ball B_{c\epsilon r}(x) of this cover,

uniformly across the cover. The normalized smooth mass averages used in the local scalar selections differ from Q by at most the same bound. Thus, after discarding finitely many selected radii, the scalar smallness and smooth-average conditions select Q on every required local ball, and its displayed budget is below d_{\mathrm{cfg},0}. Applying Lemma 2.7 with the fixed choices above now supplies on each covering ball a configuration with the prescribed majority guarantee and a configuration-distance moment whose normalized L^{p_*} norm is at most C_\epsilon\sqrt{H(r)}.

Every entry of each such configuration has size at most C_\epsilon\sqrt{H(r)}. Indeed, on a fixed majority of its base ball the entries match the actual heights within this error. The area formula, the bounded projection Jacobian, and (8.23) show that a subset of this majority has sum of squared actual heights at most C_\epsilon H(r). Every configuration entry has positive integer weight, proving the claim. Adding these configuration heights to the distance moment and summing the fixed finite cover yields

Let \mathcal I=\{t>0\} on the center. By Proposition 7.3, its relative base volume tends to zero and the height vanishes above S\setminus\mathcal I. The same is true for normalized projected mass, since Lemma 2.5 gives

Hölder’s inequality with (8.25) and (8.26) now gives

contradicting (8.24). Thus the varifold lies on the center near the origin, and Corollary 7.5 gives regularity.□

Completion of the proof of Theorem 1.1. The weighted rectifiable tangent and density theorem [19, Chapter 3, Definition 1.7, Remark 1.8, and Theorem 1.9], together with differentiation of the integer multiplicity sets, imply that at \mu-almost every point x there is a positive integer Q and an approximate tangent plane such that the dilated varifolds tend to the multiplicity-Q plane and

The tangent statement includes convergence of the tangent projections; it follows by differentiating the measurable tangent-plane map on a countable rectifiable covering. These are precisely the typical-point hypotheses used in the center construction. No regularity conclusion is used in choosing the point. Proposition 8.4 therefore shows that \mu(\operatorname{Sing}(V))=0.

For completeness, this implies the requested Hausdorff-measure conclusion even for support points outside the chosen representative of the rectifiable set. Let N\subset\spt\mu be a \mu-null set in a compact subset of U, and let O be an open neighborhood of N with arbitrarily small \mu(O). For each x\in N take arbitrarily small balls \mathbf B_r(x)\subset O\cap U. The support-point monotonicity bound in Lemma 2.1 gives \mu(\mathbf B_r(x))\geq\omega_m r^m. The disjoint-ball covering lemma selects pairwise disjoint such balls whose fivefold enlargements cover N. Their radii can all be taken below any prescribed bound, and therefore

Let \mu(O)\downarrow0 and then \epsilon\downarrow0. An exhaustion of U proves \HH^m\ll\mu on \spt\mu. Applying this to the singular set proves \HH^m(\operatorname{Sing}(V))=0.□

Proof of Corollary 1.2. For (x,P)\in G_m(S^{m+n}), define the cone by

The polar area formula [19, Chapter 3, §2, (2.4)–(2.6)] gives an integral (m+1)-varifold in \R^{m+n+1} with the same integer multiplicities away from the vertex. The vertex has zero mass, and

The radial and angular first-variation calculation below adapts the hypersurface calculation in [17, Lemma 6.1]; we include its derivation in the present dimensions. For a compactly supported C^1 ambient field X vanishing near the vertex, write X(rx)=f(r,x)x+Y_r(x) with Y_r(x)\perp x. Then

After multiplication by r^m, the radial terms integrate to the endpoint values of r^mf, and the angular term vanishes by spherical stationarity for every r. Thus the cone is stationary off the vertex. For a general compactly supported field, a radial cutoff between radii \eps and 2\eps has first-variation error O_{X,V}(\eps^{m+1}+\eps^m) by the mass formula. This tends to zero because m\geq1, so the cone is stationary everywhere.

The polar map identifies the punctured support of the cone with (0,\infty)\times\spt\|V\|. A regular link patch gives a smooth embedded local cone with the same constant multiplicity, and cone stationarity makes it minimal. Conversely, suppose the cone is locally q|M| near rx, with M smooth embedded and minimal. Its tangent planes contain the radial line almost everywhere, hence everywhere by continuity. The radial vector field is therefore tangent to M, and the radius function is a submersion on M. The dilation flow identifies a smaller patch of M with a radius interval times a smooth embedded spherical section. Comparing the polar varifold formulas on this product neighborhood identifies V there with q times that section, including its tangent planes; spherical stationarity makes the section minimal. Consequently

Apply Theorem 1.1 with dimensions (m+1,n) to obtain \HH^{m+1}(\operatorname{Sing}\mathcal C(V))=0. Let A=\operatorname{Sing}\mathcal C(V)\cap\{1<|z|<2\} and \rho(z)=|z|. Regularity is local, so A is Borel. The general Hausdorff slicing inequality [19, Chapter 2, §1, Theorem 1.10] applied to \rho gives

For 1<r<2, the slice is r\operatorname{Sing}_S V. Choosing a radius with zero slice measure and using homogeneity gives \HH^m(\operatorname{Sing}_S V)=0 in the chordal metric. The round and chordal metrics are locally bi-Lipschitz equivalent, so they have the same Hausdorff null sets. Homotheties give the assertion for every positive radius.□