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 \[\frac{\mathop{}\!d}{\mathop{}\!d\log r}\left(r^{-m}\int f(t/r)\,\mathop{}\!d\mu\right) =r^{-m}\int \frac{-f'(t/r)}{rt} |(I-P_T)(x-p)|^2\,\mathop{}\!d\mu\ge0.\] 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 (3) at every support point.

Stationarity with \(\varphi\nabla u\) shows \(\int\nabla_Tu\cdot\nabla_T\varphi\,\mathop{}\!d\mu\le0\) for every nonnegative compactly supported \(\varphi\). The preceding radial calculation, with its field multiplied by \(u\), has the additional term \[r^{-m}\int f(t/r)\nabla_Tu\cdot(x-p)\,\mathop{}\!d\mu.\] It is nonnegative: use the nonnegative radial primitive \(\varphi(t)=\int_t^\infty a f(a/r)\,\mathop{}\!da\) in the weak inequality. Thus \(r^{-m}\int_{\mathbf B_r(p)}u\,\mathop{}\!d\mu\) is nondecreasing. At integer-density points its limit is at least \(\omega_m u(p)\). Continuity and the same approximation of the center prove (4). 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 \[\begin{align*} (D\overline X)_{hh}&=J_z(\nabla X)J_z^{-1}-K_zJ_z^{-1},& (D\overline X)_{hv}v&=-A_vX,\\ (D\overline X)_{vh}a&=\mathrm{II}(J_z^{-1}a,J_zX),& (D\overline X)_{vv}&=0. \end{align*}\] The first term of the horizontal block has trace \(\mathop{\mathrm{div}}_SX\). Codazzi in Euclidean space gives \[\mathop{\mathrm{tr}}K_z=\langle\nabla_X^\perp H_S,z_{\rm phys}\rangle.\] 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 \(J_z^{-1}-I=O(|z_{\rm phys}|)\), \[\mathop{\mathrm{tr}}(D\overline X)_{hh} =\mathop{\mathrm{div}}_SX-\langle\nabla_X^\perp H_S,z_{\rm phys}\rangle +O(|z_{\rm phys}|^2|X|).\] The symmetric sum of the off-diagonal blocks, paired with \(a\in T_yS\) and \(v\in N_yS\), is exactly \[\langle a,J_z^{-1}[A_{z_{\rm phys}},A_v]X\rangle.\] 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 (6). 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 \(\int_S\mathop{\mathrm{div}}_SX\,\mathop{}\!d\mathop{\mathrm{vol}}_S=0\) proves (9).

After scaling, the coefficients of \(|\nabla X|\) and \(|X|\) are respectively \[|z_{\rm phys}|e,\qquad l\big[(|\nabla^\perp H_S|+|z_{\rm phys}|)|z_{\rm phys}| +\sqrt e\,|z_{\rm phys}|\big].\] 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 \[\det J_z=1-\langle H_S,z_{\rm phys}\rangle+O(|z_{\rm phys}|^2).\] These facts and Young’s inequality prove the bound in (11). Expanding the square root one order further gives \(\sqrt{\det(I-B)}=1-e/2+O(e^2)\), which proves (13). The rectifiable area formula (Simon 2018, chap. 3, equation (2.5)) gives the first equality in (12), 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 \[u=\sum_{ij}\partial_i\partial_j\Phi*(\chi F_{ij}) +\sum_j\partial_j\Phi*(\chi g_j)\] 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 (14) give \[\partial_jh(x)=\langle\partial_jh,\eta_x\rangle =-\sum_i\langle F_{ij},\partial_i\eta_x\rangle +\langle g_j,\eta_x\rangle +\langle u,\partial_j\eta_x\rangle.\] 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 (15), and weak convergence against tests proves (16). ◻

Proof. Scale to \(s=1\). Write \(\mathop{}\!d\mathop{\mathrm{vol}}_S=a(y)\,\mathop{}\!dy\) and put the factor \(a^{-1}\) in each coordinate component of the tangent test field. Then 4 gives an equation of the form (14) 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 6. By 5, the difference between \(a^{-1}\nu\) and \(N\,\mathop{}\!dy\) has mass at most \(C\epsilon\). The density version of this difference and (15) give \[|\{|N-c|>t\}\cap B_2|\le C\epsilon/t.\] 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 \(|c-Q|\le C\epsilon\), \[ |\{N\ne Q\}\cap B_2|\le C\epsilon. \tag{18}\] The nonnegativity of \(N\) gives \(\int_{B_2}(N-Q)_-\,\mathop{}\!dy\le Q|\{N\ne Q\}\cap B_2|\). The small measure error between \(\nu\) and \(N\mathop{\mathrm{vol}}_S\) 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 (16), applied also to the bounded volume factor, gives \(|\int\varphi\,\mathop{}\!dM|\le C\epsilon\). Hence \[M_+(B_1)\le\int\varphi\,\mathop{}\!dM+\int\varphi\,\mathop{}\!dM_- \le C\epsilon.\] This proves total variation and the exceptional-volume assertion, and rescaling proves (17). 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 \[\mathop{\mathrm{vol}}_S(\limsup_jG_j)\ge(1-\delta')\mathop{\mathrm{vol}}_S(B_s(x)).\] 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 \(C\mathop{\mathrm{Lip}}(f)l|z_{\rm phys}||X|\). In the curved case, as in 7, write \(\mathop{}\!d\mathop{\mathrm{vol}}_S=a\,\mathop{}\!dy\) and take the coordinate components \(X=a^{-1}Y\). Then \(\mathop{\mathrm{div}}_SX=a^{-1}\mathop{\mathrm{div}}Y\) exactly; derivatives of \(a^{-1}\) remain in the tensor and geometric terms. Cauchy–Schwarz, the normalized mass bound, and [lt:potential,lt:jacobian] show that the fiber function \[N_f(y)=\sum_{q\in\pi^{-1}(y),\ J_\pi(q)>0}\theta(q)f(\zeta(q))\] differs from its fixed smooth average \(\overline N_f\) by a weak-\(L^1\) quantity whose normalized size is at most \[ C\{d+s\mathop{\mathrm{Lip}}(f)\sqrt d\}. \tag{22}\] 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 7. 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 (22) 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 (18) 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 \[F(t-2w)-3\eta_0\le F_y(t)\le F(t+2w)+3\eta_0.\] 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 \[d_0(\delta_0)=\min\{d_{\rm int},d_{\rm sel}(\delta_0)\}.\] 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 [lt:scalar,lt:jacobian] bound the mass outside the configuration’s height neighborhoods by \[C(\delta_{\rm int}+d_{\rm int})r^m.\] 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 (3) 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 \[Cr\sqrt{r^{-m}\tau(\text{fixed enlargement of the parent})}.\] 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 \[\begin{align*} s^{-m}\int b_j^2\,\mathop{}\!d\mu&\le C r_j^2d, \tag{23}\\ \|b_j\|_\infty&\le C r_j \min\{1,2^{jm/2}\sqrt d\}. \tag{24}\end{align*}\] 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 \(\mathop{\mathrm{dist}}(\zeta(q),\mathcal A)\le\sum_jb_j(q)\).

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 (23) with the second part of (24), without the minimum, to obtain \[\left(s^{-m}\int b_j^p\,\mathop{}\!d\mu\right)^{1/p} \le Cs\sqrt d\,2^{-j[1-m(1/2-1/p)]}.\] Minkowski’s inequality and (19) prove (20). Summing (24) 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 (21). ◻

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 (17) on each ball and sum. This proves (25) without a factor depending on the number of balls. Apply 9 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 (26). 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 \(\dot G_s=-\mathop{\mathrm{div}}_y((y-x)G_s)\). The horizontal first variation says \(\int\mathop{\mathrm{div}}X\,\mathop{}\!dM=\int B:DX\,\mathop{}\!d\mu\). 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 (28).

For the curved identity apply 4 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 \(Q\mathop{\mathrm{vol}}_S\); its coefficient error therefore multiplies \(M\), not the background mass. The tensor coefficient errors multiply \(B\) and are bounded by \(C\gamma e\). Equation (10) controls the remaining geometric terms. Multiplying by \(s^2\) proves (29). Differentiation of \(\chi\) produces precisely the tails described in (27). ◻

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 (30) and (31). Apply the first with \(f=\Gamma\) and \(X=(y-x)G_s\). It yields the first term of (32) and the flux \(s\int G_sD\Gamma\cdot P_{vh}u\,\mathop{}\!d\mu\). Apply the normal formula with \(v=D\Gamma\) and \(\varphi=G_s\); since \(\nabla G_s=-G_su/s\), this flux is exactly \(s^2\int G_sD^2\Gamma:P_{vv}\,\mathop{}\!d\mu\). 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 \[D_h\zeta=-\omega(J_z^{-1}h)z_{\rm phys}, \qquad |D_h\zeta|\le Cl|z_{\rm phys}||h|,\] while the vertical derivative is unchanged. Differentiating \(v(\zeta)\) therefore gives the first summand of (33) 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 (33). 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 4, 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 \[Df\cdot P_{vh}(J_z\mathsf C_y-I)X_{\rm coord}, \qquad |J_z\mathsf C_y-I|\le C\gamma.\] The bound \(|P_{vh}|\le C\sqrt e\) gives its first contribution to (34). 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 \(X(\xi)\), \[\mathop{\mathrm{div}}_S(l\,DF(l\xi)X)=\mathop{\mathrm{div}}_\xi X+D_\xi(\log a_l)\cdot X, \qquad |D_\xi\log a_l|\le Cl.\] This weighted identity has no subtracted reference volume, so this last term gives the stated \(Cl\int|f||X|\,\mathop{}\!d\mu\) 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 (33) 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 (27). ◻

Proof. For \(\sigma\in\mathcal P(K)\) put \(g_\sigma=f-M_f(\pi_X)_\#\sigma\). This measure has zero mass. Define \[b_\sigma(x,s)=G_s*(-\Delta_X)^{-1}g_\sigma(x),\] 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 \[ \frac{\mathop{}\!db_\sigma}{\mathop{}\!d\log s}=-s^2(g_\sigma)_s, \qquad \partial_t b_\sigma=\Delta_X b_\sigma, \qquad t=s^2/2. \tag{42}\] 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 \[p\mathrel{% \BeginAccSupp{method=hex,unicode,ActualText=27FC}% \begingroup \let\mathrel{% \BeginAccSupp{method=hex,unicode,ActualText=21A6}% \mapsto \EndAccSupp{}}\mapsto \let\longrightarrow\longrightarrow \longmapsto \endgroup \EndAccSupp{}} \left(\frac{\exp(\beta b_p(z_{ji}))} {\sum_{k=1}^{N_j}\exp(\beta b_p(z_{jk}))}\right)_{i=1}^{N_j}\] is continuous from the probability simplex to itself. Brouwer’s Fixed Point Theorem gives a fixed point (Milnor 1978, 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 \[ \mathop{\mathrm{spt}}\sigma\subset\operatorname*{arg\,max}_{z\in K}b_\sigma(z). \tag{43}\] 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 \((x,s_0)\in\mathop{\mathrm{spt}}\sigma\), (43) 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 (42) proves the first inequality in (41); integrating that identity proves the second. ◻

Proof. The only commutator is \[f_s(x)=w(x)L'_s(x)-\eta\nu_s(x)+E_s(x),\qquad |E_s(x)|\le C_0\int G_s(x-y)|w(y)-w(x)|\,\mathop{}\!d\tau(y).\] 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 \[C\tau(X)s^{-m}\exp(-c_2\eta^2R^2/s^2).\] 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 (45).

At a contact \(f_{s_0}\ge h_{s_0}\ge0\). If \(w(x)=0\), (45) would give a strictly negative value. Division by \(w(x)>0\), followed by \(w(x)\le1\), proves (46). Integrate (45), 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 (47). ◻

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 \[b(T)=1+(\log T)^{201} +4F_{Q-1}^{-1}((\log T)^3).\] The definition of the iterated exponentials gives \(F_{Q-1}^{-1}((\log T)^3)=T^{o(1)}\), and this choice has (50). 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 \[ J_0>4N(m+2)+100, \tag{52}\] and enlarge \(J_0\) to exceed all fixed polynomial density losses in the cell constructions below. Set \[ \eta=T^{-N},\quad \varepsilon=\frac{(\log T)^{60}}{T},\quad r_b=\sqrt d\,e^{3b},\quad n_0=\lceil10N\log T\rceil, \quad E_j(t)=T^{J_0(j+1)}d\,t^{-2-\varepsilon}. \tag{53}\] 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 \[ \log E_{2n_0+4}(r_b/T^4) =-6b+\tfrac12\ell_T^{60}-3\varepsilon b +O_{J_0,N}(\ell_T^2) <-2b. \tag{54}\] Also, for \(i\le n_0\), \[ v\sqrt{E_{2i+1}(v)} =\sqrt d\,T^{J_0(i+1)}v^{-\varepsilon/2}. \tag{55}\] 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 \[ \tau(B_{4R}(x_*))\le C R^mE_{2i}(R),\qquad R\ge cr_b/T^2, \tag{56}\] 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 \[d_\infty(\mathcal A_R,\mathcal A)\le(i+1)e^b\sqrt d.\] 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 \(0\le w\le1\), \(\mathop{\mathrm{Lip}}w\le R^{-1}\), supported in \(B_R(x_*)\). Write \[L'=L-C_Q\Lambda+c(T)D_{\rm far},\qquad \mathcal N_R=\tau(B_{4R}(x_*))+dR^m.\] We prove, by downward induction on \(i\), that \[ \int w\,\mathop{}\!dL'\le p_i\mathcal N_R, \qquad p_{n_0}=C_*,\qquad p_i=C\eta+\tfrac12p_{i+1}. \tag{57}\] 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.

Proof. Only the first assertion needs proof. Put \[\beta_\sigma=(\pi_X)_\# \bigl(\widehat h_\sigma\!\restriction_{K_\sigma\setminus\mathcal G_\sigma}\bigr) \le h_\sigma,\] 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 \[h_\sigma(J)\le C\sigma^m(f_\sigma(x_J))_+ \le Cd_*\sigma^m.\] Since \(\beta_\sigma\le h_\sigma\), this also bounds \(\beta_\sigma(J)\). On an occupied half-open grid cube define \[g_\sigma|_J=C\sup_{x\in\mathcal C_\sigma\cap J} (f_\sigma(x))_+, \qquad \overline\beta_\sigma =\sum_J\frac{\beta_\sigma(J)}{|J|}\mathbf1_J\,\mathop{}\!dy,\] and put \(g_\sigma=0\) on unoccupied cubes. Then \(\overline\beta_\sigma\le g_\sigma\,\mathop{}\!dy\) and \(0\le g_\sigma\le Cd_*\). For every continuous test \(\varphi\), \[\left|\int\varphi\,\mathop{}\!d\beta_\sigma -\int\varphi\,\mathop{}\!d\overline\beta_\sigma\right| \le\omega_\varphi(C\sigma)\beta_\sigma(\text{chart})\longrightarrow0,\] 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 \[\sup_{|x-y|\le C\sigma} \big|(G_\sigma*\lambda)(x)-\lambda_{\mathrm{ac}}(y)\big| \longrightarrow0.\] 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 \[\limsup_{\sigma\downarrow0}g_\sigma(y) \le C\mathbf1_{\{\tau_{\mathrm{ac}}\le d_*\}} (f_{\mathrm{ac}})_+(y) \quad\text{almost everywhere}.\] 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 \[ C\,\mathbf 1_{\{\tau_{\mathrm{ac}}\le Cd_*\}} (f_{\mathrm{ac}})_+. \tag{85}\] The value of the fixed constant in the threshold is immaterial.

At a point in this set, the area formula and 5 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 \[2(1-J_\pi)-e \le Ce^2+C\bigl(|z_{\mathrm{phys}}|^2+|H_S|^2\bigr).\] 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 (85). Its integral is at most \(Cd_*d\). Compactness and (85) prove the stated uniform limsup bound, hence the assertion. ◻

Proof. Write \(I_{\mathrm{end}}\) for the artificial term in (88). The definition makes \(I\) positive and continuous. Let \[J_\tau(s)=\sup_{s\le v\le\rho}(s/v)^K v^2\tau_v.\] 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 \[ e^{-T/2}\le\delta\le d_*,\qquad \tau_t\le(t/v)^{K-2}\delta\quad(v\le t\le\rho). \tag{92}\] 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 \[\frac{J_\tau(s)}{I_{\mathrm{end}}(s)} \le C\rho^{m+4}T^{N(K+m-2)}=o(1),\] 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 (92) imply \[ r^{-m}\tau(B_{4r}(y)) \le C(1+R)^{2m+K-2}\delta. \tag{93}\] For \(R\le C\sqrt{|\log\delta|}\) this is small, uniformly in \(\delta\le d_*\). Moreover \(\tau\) includes \(|M|=|\pi_\#\mu-Q\mathop{\mathrm{vol}}_S|\), so the normalized smooth mass average on each cell is close to that same \(Q\). The integer in 7 is therefore \(Q\) on every cell. The Gaussian has a bounded ratio on a fixed enlargement of such a cell, and 10 sums the estimates with bounded overlap.

Annular summation of (92) 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 \[|M|_v\le C(D_v+\Lambda_v)+o(\tau_v),\qquad \tau_v\le C(D_v+\Lambda_v).\]

Suppose \(A(v)\le\varepsilon v^2\tau_v\). For \(C_Q\) sufficiently large and then \(\varepsilon\) sufficiently small, \[(f_*)_v\le-c\tau_v.\] 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 (87), \[Z(e^\vartheta v) \le Z(v)-c'\vartheta v^2\tau_v \le \varepsilon v^2\tau_v-c'\vartheta v^2\tau_v<0\] 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 (89). 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 (91) 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 \[a(\rho)=T+(2m+4)\log\rho=T-o(T)>T/2.\] 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 11 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 (90) 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 7. Choose a joint majority-and-moment witness \(\mathcal A_0=\{a_1,\ldots,a_Q\}\) from 9 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 10, applied only on this body, and \(\Gamma_{s_0}\le C\mathop{\mathrm{dist}}(z,\mathcal A_0)^2\) give \[\int_{|u|\le R_0}G_{s_0}\Gamma_{s_0}\,\mathop{}\!d\mu \le Cs_0^2d_*.\] The root mass identity and threshold control at \(\sqrt2s_0<\rho\) give \[(\pi_\#\mu)_{\sqrt2s_0} \le C+\tau_{\sqrt2s_0} +Cs_0^{-m}e^{-c/s_0^2}\le C.\] The last term accounts for the fixed exterior-chart and periodic tails. Since \(\Gamma_{s_0}\le Cs_0^2\), Gaussian comparison yields \[\int_{|u|>R_0}G_{s_0}\Gamma_{s_0}\,\mathop{}\!d\mu \le Cs_0^2e^{-R_0^2/4}(\pi_\#\mu)_{\sqrt2s_0} \le Cs_0^2e^{-C_b^2a_0/4}=o(s_0^2d_*).\] 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, \[ \begin{split} r^{-m}\tau(B_{4r}(y))&\le d_c:=C(1+R_y)^{2m}d_*,\\ T_c:=-\log d_c&=a_0-2m\log(1+R_y)-O(1),\\ r\sqrt{d_c}&\le Cv(1+R_y)^{m-1}\sqrt{d_*}. \end{split} \tag{97}\] 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 7 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 16. Multiply its heights by \(r\) when comparing them in the fixed coordinates of this section. The overlap and matching rules (48)–(49) 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 16. 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 16 is nonnegative. The resulting bound, integrated in scale, gives \[Z(s_0)\le Ca_0^{-P-1}d_* \int_{\underline s}^{s_0}v^2\,\mathop{}\!d\log v \le a_0^{-P}I(s_0).\] 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 (87). 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 \[A(\underline s)+\int_{\underline s}^{s_0}v^2R_v\,\mathop{}\!d\log v \ge cI(s_0).\] 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 (84) 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 10 directly on this earlier interval, using the scalar hypotheses verified in (97). 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 10 to joint configuration witnesses on the admissible dilations and match their centers to the common well. Gaussian summation absorbs their polynomial losses and gives \[\int G_v|V_v|^2\,\mathop{}\!d\mu \le Cv^2S_h(a_0)^2d_*.\] The support estimates for those chosen witnesses put all support heights on \(|y-x|\le C_b\sqrt{a_0}v\) within \[v\bigl[\operatorname{poly}(a_0)e^{-\beta a_0} +S_h(a_0)e^{-a_0/2}\bigr]=o(v)\] of the common well. Clipping is therefore absent on this body. For the clipped tail, \(\pi_\#\mu=Q\mathop{\mathrm{vol}}_S+M\) 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 12 now gives \[D_v\le \frac1v \left|\int G_v V_v\cdot P_{vh}u\,\mathop{}\!d\mu\right|+o(d_*).\] 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 \[c v^2d_*^2 \le \left(\int G_v|V_v|^2\,\mathop{}\!d\mu\right)R_v \le Cv^2S_h(a_0)^2d_*R_v, \qquad v^2R_v\ge cS_h(a_0)^{-2}v^2d_*.\] 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 (87). 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 \[ g\le s\min\{C_j e^{-\gamma a},\, S_1(a)e^{-a/2}k^{-1}\}. \tag{102}\] 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 \[ k_{\mathrm{new}}=c k^6S_1(a)^{-6}. \tag{103}\] 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 (98) on a stage starting at level \(a_j\) is \[k(a)^{-1}=k(a_j)^{-1}+a_j^{-1}-a^{-1}.\] 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 \[\log(1/k)\le C_Q\{1+b(T)+b(a_0)\}=o(a_0),\] where 21 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 9 on the balls under consideration and match their centers to these lists. The support and matching bounds, together with (89), 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, \[ \mathop{\mathrm{dist}}(z,\mathcal W)\le Cs e^{-\gamma' a} \quad\text{on }|y-x|/s\le C\sqrt a \tag{104}\] 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 \[\max_{\xi\in\mathcal A_s}\Gamma_s(\xi) \le C U+Cs^2e^{-a} \le Cs^2e^{-a}/k.\] Since \(\Gamma_s\) is comparable to \(\min\{\mathop{\mathrm{dist}}(z,\mathcal W)^2,s^2\}\) 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 (104) 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 (48), their lists match by (49), 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 16. 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 \[ \begin{split} r^{-m}\tau(B_{4r}(y)) &\le d_c:=C(1+R_y)^{2m+K-2}e^{-a},\\ T_c:=-\log d_c &=a-(2m+K-2)\log(1+R_y)-O(1),\\ r\sqrt{d_c} &\le Cs(1+R_y)^{m+K/2-2}e^{-a/2}. \end{split} \tag{105}\] The cell and its fixed enlargements lie in the chart, and \(T_c=a-O(\log a)\to\infty\). As in (97), 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 (89); 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 (102) 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 16 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 (101).

We prove the two differential inequalities explicitly. In flat coordinates the horizontal test with multiplier \(\Gamma_s\) gives \[\dot U=\int G_s\{\Gamma_s(B[u,u]-e) +sD\Gamma_s\cdot P_{vh}u+\dot\Gamma_s\}\,\mathop{}\!d\mu, \qquad u=(y-x)/s.\] Normal stationarity with \(G_sD\Gamma_s\) converts the cross term to \(s^2\int G_sD^2\Gamma_s:P_{vv}\,\mathop{}\!d\mu\). On the pure regions \(D^2\Gamma_s=2\operatorname{Id}\), and elsewhere its norm is bounded. Dropping \(-\Gamma_se\), and using (104) to bound the \(\Gamma_s B[u,u]\) term, gives (99). 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 (104).

For the second inequality use \(|P_{vh}u|^2\le B[u,u]\) and complete the square: \[s^2R_s\ge 4ks\int G_sV_b\cdot P_{vh}u\,\mathop{}\!d\mu-4k^2U.\] Normal stationarity gives \[s\int G_sV_b\cdot P_{vh}u\,\mathop{}\!d\mu =s^2\int G_sDV_b:P_{vv}\,\mathop{}\!d\mu \ge s^2D_s-Cs^2D_{\mathrm{tr}}\] up to the already specified tails. This proves (100) 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 (89) directly. No inverse power of \(k\) is required for these flat errors, because the derivative bounds in (95) are uniform. For the curved errors, keep the original coordinates scaled by the physical radius \(l\), and write \(\langle f\rangle_s=\int G_sf\,\mathop{}\!d\mu\) 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 12 satisfy \[\begin{align*} s^2l\langle|z_{\mathrm{phys}}|\sqrt e\rangle_s &\le s^2l\sqrt{\Lambda_sD_s}\le CP(a)lI,\\ s^2l\langle|v|(|H_S|+|z_{\mathrm{phys}}|+e+\sqrt e)\rangle_s &\le Cs^2l\sqrt U\sqrt{\Lambda_s+D_s} \le CP(a)ls k^{-1/2}I,\\ \gamma_{\rm geom} s\langle|u||v|\sqrt e\rangle_s &\le\gamma_{\rm geom} s\sqrt U\sqrt{\langle|u|^2e\rangle_s} \le CP(a)\gamma_{\rm geom} k^{-1/2}I. \end{align*}\] 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 \[\gamma_{\rm geom} s\langle|u||D\Gamma_s|(\sqrt e+\sqrt\lambda)\rangle_s \le C\gamma_{\rm geom} s\sqrt U \sqrt{\langle|u|^2(e+\lambda)\rangle_s} \le C\gamma_{\rm geom} P(a)k^{-1/2}I.\] 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 \[\delta_\alpha=\alpha_a-q\alpha,\qquad h_1=q+\delta_\alpha Z/I.\] Differentiating \(I-kU-\alpha Z\), and using \(\dot a=2-\dot I/I\), gives \[ (I-kU-\alpha Z)^{\boldsymbol\cdot} =(1+h_1)\dot I-2h_1I-k\dot U-\alpha\dot Z. \tag{106}\] At a reset write \(\mathcal D=s^2D_s\), \(\mathcal R=s^2R_s\). Apart from the absorbed geometric error, \[\dot A=2I-\mathcal D+\mathcal R,\qquad \mathcal D=2I-\dot Z-J.\] Insert (99) and (100) into (106) and use \(kU=I-\alpha Z\). The resulting lower bound is \[\begin{align*} &(4k-2)h_1 I+4k(1+h_1)\alpha Z +c_Z\dot Z-Cks^2D_{\mathrm{tr}}-O(\mathcal E kI) +c_JJ,\tag{107}\\ &c_Z=1+h_1-2k-4kh_1-\alpha,\qquad c_J=2k-(4k-1)(1+h_1). \end{align*}\] 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, \[\delta_\alpha=\frac{k}{a^2} +\frac{2k(1-4k)}{a^3}-\frac{4k^3}{a^4}=O(k/a^2),\qquad c_Z=(1-4k)\delta_\alpha Z/I.\] 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 (107) is at least \(ck\). Hence the reset derivative is strictly positive for large \(a\).

With several wells there is the useful exact cancellation \[\delta_\alpha=\frac{k}{a^2} +C_1k\frac{\log a-1}{(\log a)^2}.\] In particular \(\delta_\alpha=o(k)\), \(|h_1|\le C/a\), and \[-C C_1ka/\log a\le c_Z\le-c C_1ka/\log a.\] Multiply (101) by this negative coefficient. It supplies a favorable multiple of \(C_1k\log a\,s^2D_{\mathrm{tr}}\), which absorbs the transition loss in (107). 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 \[(I-kU-\alpha Z)^{\boldsymbol\cdot} \ge\{(1+h_1)K-2h_1-2Ck\}I +(Ck-\alpha)\dot Z-O(\mathcal E kI).\] The coefficient \(Ck-\alpha\) is negative. A negative \(\dot Z\) therefore helps. For one well, \(\dot Z\le2I\); for several wells, (101) 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 19; 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 \[U_{\mathrm{new}}\le C(U+g^2),\qquad g^2\le S_1(a)^2k^{-2}I.\] 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 (103), \(\theta\ll1\) and \(\lambda\ll k\). The old comparison then gives \[k_{\mathrm{new}}U_{\mathrm{new}}+\alpha_{\mathrm{new}}Z \le(\theta+\lambda)I+ (\alpha_{\mathrm{new}}-\theta\alpha)Z\le I.\] 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, \[ F_{Q-1}^{-1}(u) =\exp_j\bigl((\log_j u)^2\bigr). \tag{109}\] 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 \[\exp\!\left(\bigl[F_1(c\log u)\bigr]^{1/3}\right).\] It follows directly from (109) that, for some fixed \(\varsigma>0\), every such inverse dominates \[\exp_{2Q-1}\bigl((\log_{2Q-1}u)^{\varsigma}\bigr).\] 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 (103); 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 (108). 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 (103), 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 (108) and the explicit solution of (98) then give \(k\ge B_i^{-\varepsilon_0}\) 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 [ex:envelope,ex:barrier], \(0\le Z\le A\le I\), \(A'-Z'\ge0\), and \[ |a'|+|I'|/I+|Z'|/I\le C,\qquad \left|(Z'/I)'\right|\le C \tag{113}\] 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 \[ |M|_s\le C_0(D_s+\Lambda_s)+a^{-P}I/s^2, \tag{114}\] where \(P\) can be chosen as large as needed. With several wells, \[ Z'+c_0\frac{(\log a)^2}{a}s^2D_{\mathrm{tr}}(s) \le a^{-P}I. \tag{115}\] Here \(D_{\mathrm{tr}}(s)\) is the Gaussian average over the enlarged transition set of 20. 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 \[n\le a(t)<n+1,\qquad e^{-j-1}M_*<I(t)\le e^{-j}M_*\] confine \(t\) to an interval of length at most one, because \[ t=\frac{a(t)+\log I(t)}2. \tag{116}\] 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 \[ \int_{t\le t_1}a(t)^{-P}I(t)\,\mathop{}\!dt \le CM_*\sum_{j\ge0}e^{-j} \sum_{n\ge\lfloor A_*\rfloor}n^{-P} \le C A_*^{1-P}M_*. \tag{117}\] 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 (115) and (117), 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 \[A(t)\ge A(u)-Z(u) \ge A(u)-Z(t_0)-\int_{t_0}^t(Z')_+\,\mathop{}\!dv.\] The future envelope has the elementary property \[ \max_{t_0\le u\le t}I(u) \le\max\left\{I(t),\max_{t_0\le v\le t}A(v)\right\}. \tag{118}\] 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 \[ I(t)\ge \tfrac12\max_{t_0\le u\le t}I(u). \tag{119}\] This argument is unchanged at a merger: \(I,A,Z\) do not jump, and (115) 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 \[ \int_{t_b}^t(Z')_+\,\mathop{}\!dv\le CB^{1-P}M(t),\qquad Z(t)\le I_b+CB^{1-P}M(t),\qquad I(t)\ge M(t)/2. \tag{120}\] 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 (117) on each piece shows that \[\int\frac{(Z')_+}{M}\,\mathop{}\!dt\le CB^{1-P}\log B.\] Integration by parts against the continuous nondecreasing function \(M\) gives \[\int\frac{-Z'}M\,\mathop{}\!dt =\frac{Z(t_b)}{M(t_b)}-\frac{Z(t)}{M(t)} -\int \frac{Z}{M^2}\,\mathop{}\!dM\le1.\] Consequently \(\int (Z')_-/M\le2\) for large \(B\). In view of (119), 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 \[ C B/(\log B)^2=o(B). \tag{121}\] Constants here may depend on the fixed \(\delta_1\).

After this growth of \(M\), (120) implies \(Z(t)\le C(B^{-10}+B^{1-P})I(t)\). If \(Z'/I<-h_B\) at a time with \(B+1<a<2B-1\), (113) 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 (121).

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 \[C_Q\Lambda_s\le(2+h_B)e^{-a}.\] Apply (114). Taking \(C_Q\) sufficiently large compared with \(C_0\), and then \(B\) large, forces \(D_s\ge c e^{-a}\). Moreover (115) gives \[D_{\mathrm{tr}}(s) \le C\left(\delta_1+\frac{B^{1-P}}{(\log B)^2}\right)e^{-a}.\] Choose \(\delta_1\) first so that the last bound is at most \(\vartheta e^{-a}\). This proves (111) 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 (112), then throughout it \(s^2D_s<B^{-\varepsilon_0}I\). At a reset the Gaussian horizontal identity gives \[I'=2I-s^2D_s+s^2R_s+o(I)\ge I.\] Also (114), 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 \[Z'\le C s^2D_s+o(I)\le C B^{-\varepsilon_0}I.\] 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 \(W'\le C B^{-\varepsilon_0}\). If the two time sets have lengths \(\mathcal R,\mathcal N\), integration gives \[\mathcal R\le1+C B^{-\varepsilon_0}\mathcal N.\] On the other hand \(a'\le1\) at resets, while \(a'=2-K\) at nonresets. A net increase of \(B^{1/4}\) therefore requires \[\mathcal R\ge B^{1/4}+(K-2)\mathcal N,\] 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 (110) 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 (Hytönen and Kairema 2012, 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 (122) has room to spare.

Let \(V_b\) be the normal multiplier of 20, 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 20 by \(s^2\), with its Gaussian cutoff, therefore gives \[ \mathcal F:=\int G_s\,v\cdot P_{vh}u\,\mathop{}\!d\mu \ge c B^{-\varepsilon_0}\delta. \tag{124}\] For multiple wells the discarded terms are bounded by a fixed multiple of \(D_{\mathrm{tr}}(s)\), and \(\vartheta\) in 22 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 20, and the curved first-variation errors are \(o(B^{-\varepsilon_0}\delta)\). This last assertion follows from \(k\ge B^{-\varepsilon_0}\), the moment barrier, the dilation bound, and the physical exponential smallness in 13.

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 (19), 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 \[(Rs)^{-m}\tau(B_{cRs}(x)) \le d_R:=C(1+R)^{K-2}\delta=o(1)\] on the required fixed enlargement. Its \(|M|\) part identifies \(Q\) by the smooth mass average in 7. Choose the joint majority-and-\(L^p\) witness \(\mathcal B_R\) in 9, 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 \[\left((Rs)^{-m}\int_{\pi^{-1}(B_{Rs}(x))} \mathop{\mathrm{dist}}(z,\mathcal B_R)^p\,\mathop{}\!d\mu\right)^{1/p} \le C(1+R)^{K/2}s\sqrt\delta.\] 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 \[\max_{\xi\in\mathcal B_1}|V_b(\xi)|^2 \le C U+Cs^2\delta \le Ck^{-1}s^2\delta.\] 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, \[|v(z/s)|\le \frac C s\mathop{\mathrm{dist}}(z,\mathcal B_R) +\max_{\xi\in\mathcal B_R}\frac{|V_b(\xi)|}{s}.\] The last three bounds and the bounded normalized projected mass give \[ \left((Rs)^{-m}\int_{\pi^{-1}(B_{Rs}(x))}|v|^p\,\mathop{}\!d\mu \right)^{1/p} \le C(1+R)^C k^{-1/2}\sqrt\delta. \tag{125}\] All these bounds use displacements from the fixed wells. Gaussian summation of (125) and of the energy dilation bound gives \[ \left(\int G_s|v|^p\,\mathop{}\!d\mu\right)^{1/p} \le B^{c_1\varepsilon_0}\sqrt\delta, \qquad \int G_s(1+|u|^2)e\,\mathop{}\!d\mu\le C\delta. \tag{126}\] 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 \[\int_{|u|>R}G_s|v|^p\,\mathop{}\!d\mu\le Ce^{-R^2/4}.\] 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 (125) makes both the flux tail outside \(B_{R_Bs}(x)\) and the tilt tail there smaller than \(o(B^{-\varepsilon_0}\delta)\). Choose \(J\) to contain this ball in its interior, with \(\ell(J)\le C R_Bs\). The remaining tilt has Gaussian integral at least \(cB^{-\varepsilon_0}\delta\). Since \(G_s\le Cs^{-m}\), \[\nu(J)\ge cB^{-\varepsilon_0}\delta s^m.\] 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 \(B^{\varepsilon_0}\) after increasing the threshold for \(B\). This proves (122).

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 (122) gives \[ \frac{|H|}{|J|}\le B^{\varepsilon_0-C_*}. \tag{127}\] It is essential here to estimate mass as well as volume. The identity \(\pi_\#\mu=Q\mathop{\mathrm{vol}}_S+M\) gives \[ \frac{\mu(\pi^{-1}H)}{|J|} \le C Q\frac{|H|}{|J|}+\frac{|M|(J)}{|J|} \le C B^{\varepsilon_0-C_*}+B^{\varepsilon_0}\delta. \tag{128}\] 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 \(B^{c_2\varepsilon_0-C_*}+B^{c_2\varepsilon_0}\delta\), after absorbing the volume ratio \(|J|/s^m\) into a power of \(B^{\varepsilon_0}\). Hölder’s inequality with exponents \(p,2,1/\sigma\), together with \(|P_{vh}u|\le\sqrt e\,|u|\) and (126), bounds the absolute flux on this set by \[ B^{c_3\varepsilon_0}\delta \left(B^{c_2\varepsilon_0-C_*}+B^{c_2\varepsilon_0}\delta\right)^\sigma. \tag{129}\] Choose \(C_*\) large, independently of sufficiently small \(\varepsilon_0\), and then \(B\) large. Expression (129) is less than one quarter of the lower bound in (124). The strict inequality \(p>2\) is used exactly at this step.

There remains flux at least \(cB^{-\varepsilon_0}\delta\) over \((J\setminus H)\cap B_{R_Bs}(x)\). Cauchy–Schwarz, the second-moment consequence of (126), and \(|u|^2\le R_B^2\) on this core imply \[cB^{-2\varepsilon_0}\delta^2 \le B^{c_4\varepsilon_0}\delta\, C R_B^2s^{-m}\nu(J\setminus H).\] Thus \(\nu(J\setminus H)\ge B^{-c_5\varepsilon_0}\delta s^m\). Compare with \(\nu(J)\le B^{\varepsilon_0}\delta|J|\) and absorb the remaining fixed logarithmic factors. This proves (123) 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 \[J_*=1+\left\lfloor\log(\lambda_+/\lambda_-)\right\rfloor, \qquad \lambda_j=\lambda_-e^j\quad(0\le j<J_*).\] 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 \[\sum_R\nu(R) \le \frac{e\lambda_j}{R_*}\sum_R|R| \le\frac{e\lambda_j}{R_*}|H| \le\frac e{R_*}\nu(H).\] 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 \[ \nu(E_{\mathrm{ret}})\le(eJ_*/R_*)\nu(X). \tag{133}\] 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 (130), \[\sum_C\nu(C)\le(1-\zeta)\nu(P).\] 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 \[ \nu(\text{union of paying generation }j) \le(1-\zeta)^{j-1}\nu(X). \tag{134}\]

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 \(j\)th 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 \[d(W_{j+1})\ge S d(W_j)>L d(P_j).\] 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 \[E_N\subset E_{\mathrm{ret}} \cup(\text{union of paying generation }N).\] Equations (133) and (134) prove (131). 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 24 to all cubes which can occur in 23, 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 (130) with the uniform choices \[\begin{aligned} \lambda_-&=cB^{-2\varepsilon_0}e^{-2B},& \lambda_+&=B^{\varepsilon_0}e^{-B},\\ L&=\max\{1,c^{-1}B^{C_*+2\varepsilon_0}\},& \zeta&=B^{-c_*\varepsilon_0}. \end{aligned}\] 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 (113) and (110), consecutive radii differ by at least \(\exp(cB^{1/3})\). The first two properties in (122) 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 (131) is at most \[\left[\exp\{-cB^{1/2-c_*\varepsilon_0}\} +CB\exp(-B^{1/4})\right]d.\] Choose \(\varepsilon_0\) so small that \(c_*\varepsilon_0<1/4\). This proves (132) 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 \[G_{s_x}(y-x)\le 2^{m/2}e^{-L_0^2/4} G_{\sqrt2s_x}(y-x).\] Choose \(L_0\) sufficiently large. By (135), 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 \[ \nu(B_x)\ge c\theta s_x^m. \tag{137}\]

Apply the usual decreasing-radius selection to these balls; see (Simon 2018, chap. 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 \[|x-x_i|<3R_i,\qquad u_x\le s_x\le2R_i/L_0.\] 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 \(c\theta\le\nu_{s_x}(x)\le C_m s_x^{-m}\nu(\mathbb R^m)\).

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 (135) give \[h(5D_{ij})\le C\theta u_{ij}^m.\] 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 (137). The \(B_i\) are disjoint and contained in \(E\), which proves (136). 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 13. Fix \(Q\) and assume the assertion for all smaller positive integers. Choose \(C_Q\) large enough for [ct:separated,ex:envelope,ex:barrier,pk:witnesses] and the bottom-contact estimate in 17. 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 18. 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 5. Write \(\widehat h=\widehat h_\sigma\) and \(h=h_\sigma=(\pi_X)_\#\widehat h\); both have mass \(\int f\). By 17, the \(\widehat h\)-mass of \(K_\sigma\setminus\mathcal G_\sigma\) is at most \[ o(1)+Ce^{-a_0}d \tag{138}\] 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 \[\tau_{s_0}(x)=e^{-a_0},\qquad \tau_v(x)\le e^{-a_0}\quad(\underline s\le v\le\rho), \qquad h_{\underline s}(x)\le Ce^{-a_0}.\] 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 (135) 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 [pk:witnesses,pk:reverse]. 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 24 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 \[\log(s_{\min}/s_0)\ge c(B_i-a_0-O(1))\longrightarrow\infty.\] The cube contains \(B_{s_{\min}}(x)\), so it contains \(B_{L_0s_0}(x)\) for the fixed constant in 25.

For every subset \(A\subset K_\sigma\), the induced outer measures obey \[\widehat h^*(A)\le h^*(\pi_X A).\] 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 25 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 \[\begin{align*} \widehat h(\mathcal G_\sigma) &=\widehat h(\mathcal G_\sigma\cap\mathcal S_\sigma)\\ &\le\sum_{i,g}\widehat h^*(A_{i,g}) \le\sum_{i,g}h^*(\pi_X A_{i,g})\\ &\le C\sum_{i,g}\nu(E_{i,g}) \le C\sum_{i=1}^Q e^{-B_i^{1/8}}d. \tag{139}\end{align*}\]

The band choice in 21 ensures \(B_1^{1/10}\gg F^3\). Consequently the right side of (139), 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 (138). 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 \[\int w(L-C_Q\Lambda)\le C e^{-2F_Q(T)}d \le e^{-F_Q(T)}d\] for sufficiently large \(T\), uniformly over the stated nonnegative unit Lipschitz weights. This proves the induction step and hence 13 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 \[ \mathop{\mathrm{tr}}(P\,DZ) =e-P_{hh}:A_z(I-A_z)^{-1} =e-H_{S'}\cdot z+O(|A|^2|z|^2+|A|e|z|). \tag{148}\] Testing stationarity with \(\chi^2Z\), as in the classical height-to-tilt estimate (Allard 1987, Lemma 31), absorbing the last term, and using \(|P_{vh}|^2\le e\) gives \[\int\chi^2e\,\mathop{}\!d\mu \le C\int\bigl(|\nabla\chi|^2+|A|^2\chi^2\bigr)|z|^2\,\mathop{}\!d\mu \le Ca^2.\] Here \(H_{S'}=0\) is essential: it removes the term linear in height. 7 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 7 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 \(\mathop{\mathrm{vol}}_{S'}\). Normal stationarity and horizontal stationarity with a linear height multiplier imply \[ \left|\int_{S'}\left( \nabla^\perp u:\nabla^\perp v-\mathop{\mathrm{tr}}(A_uA_v)\right) \mathop{}\!d\mathop{\mathrm{vol}}_{S'}\right| \le Ca^2\|v\|_{C^N} \tag{149}\] 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 \[-P_{hh}:A_v(I-A_z)^{-1} +P_{vh}:\nabla^\perp v\,(I-A_z)^{-1}.\] Its terms of first order in \((z,P_{vh})\) are \(-\mathop{\mathrm{tr}}(A_zA_v)+P_{vh}:\nabla^\perp v\); 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 (149), 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 (149) into an exact minimal correction. Choose an integer \(q>m/2+1\) sufficiently large that the right side of (149) 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 \[ \|\mathcal RF\|_{H^{t+2}} \le C_t\|F\|_{H^t},\qquad -q-2\le t\le s, \qquad \mathcal J\mathcal RF=F \quad\hbox{on the smaller ball}. \tag{150}\] 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 (150) 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 (150): 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 \[\|\mathcal N(v)\|_{H^s}\le C\|v\|_{H^{s+2}}^2, \quad \|\mathcal N(v)-\mathcal N(w)\|_{H^s} \le C(\|v\|_{H^{s+2}}+\|w\|_{H^{s+2}}) \|v-w\|_{H^{s+2}}.\] 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 \(Q\mathop{\mathrm{vol}}_{S'}\) costs \(O(a^3)\), by the mass-defect bound and its \(O(a)\) size. The first-order term is \(Q(u-v)\), proving (146). 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 7 gives its mass estimate.

We apply this procedure at \(j=0\) using the horizontal plane as input, with initial errors small enough that (144) 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 (149), then \(q,s\) and the finite coefficient order required by (150), 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 7 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 \(\int d\zeta\,\mathop{}\!d((\pi_0)_\#\mu)=O(a^2)\|\zeta\|_{H^q}\), 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 7 changes the measure to \(Q\mathop{\mathrm{vol}}_{S_1}\) 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 (147). ◻

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 \(B_{70}\), \[ a^{-2}\pi_\#(|z|^2\mu)\rightharpoonup\nu, \qquad a^{-2}\pi_\#(B\mu)\rightharpoonup\beta, \qquad a^{-2}D\rightharpoonup d, \qquad a^{-2}M\rightharpoonup\mathfrak m, \qquad d=\mathop{\mathrm{tr}}\beta. \tag{151}\] 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 \(\pi_\#\mu=Q\mathop{\mathrm{vol}}_S+O(a^2)\) in total variation imply \(\nu\le C\,\mathop{}\!dy\) 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 (148) 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 3 shows that \(\nu(B_{70})\ge c>0\).

The signed estimate gives the crucial additional inequality \[ 2\mathfrak m\le d. \tag{152}\] We check its physical-scale hypothesis even if \(\ell\) stays bounded away from zero. Before applying 13, 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 \[ 0<\kappa<\min\left\{\frac\sigma2, \frac1{2(k_0-1)}\right\}. \tag{153}\] 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 13 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 (152).

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 \[\phi_s(y)=s^{-m}\left(1-\frac{|y-y_0|^2}{s^2}\right)_+^p, \qquad \psi_s(y)=2p\,s^{-m}\left(1-\frac{|y-y_0|^2}{s^2}\right)_+^{p-1}.\] In this paragraph only, define \[\mathcal H=\int\psi_s\,\mathop{}\!d\nu, \quad \mathcal A=s^2\int\phi_s\,\mathop{}\!d\mathfrak m, \quad \mathcal D=s^2\int\phi_s\,\mathop{}\!dd, \quad \mathcal R=\int\psi_s\,\mathop{}\!d\beta[y-y_0,y-y_0].\] Dots denote differentiation in \(\log s\). Stationarity gives \[ \dot{\mathcal A}=2\mathcal A-\mathcal D+\mathcal R, \qquad \dot{\mathcal H}=2\mathcal D, \qquad \mathcal R\mathcal H\ge\mathcal D^2, \qquad 2\mathcal A\le\mathcal D. \tag{154}\] 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 \(s^2\int\phi_s e/a^2\,\mathop{}\!d\mu\) with \[\int\psi_s\,\frac{z\cdot P_{vh}(y-y_0)}{a^2}\,\mathop{}\!d\mu+o(1).\] 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 (148) and 4. 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 (152).

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 \[ n\ge0,\qquad 2P\le n,\qquad \dot P\ge 2P-n+n^2-2nP. \tag{155}\] 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 (154), applied also to smaller balls and other centers, gives \(d=0\). To pass the total-variation estimate of 7 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 \[|\mathfrak m(\zeta)| \le C\lim_i\int\chi\,a_i^{-2}(D_i+\Lambda_i) =C\int\chi\,\mathop{}\!dd=0.\] 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 \[(e^{-2t}\mathcal A)^{\displaystyle\cdot} =e^{-2t}(\mathcal R-\mathcal D) \ge-\tfrac14e^{-2t}\mathcal H,\] because \(n^2-n\ge-1/4\). The function \(\mathcal H\) is nondecreasing. Integrating from \(t_0\) therefore gives the explicit lower bound \[P(t)\ge-\frac{e^{2(t-t_0)}-1}{8}.\] With \(P\) bounded on this finite interval, (155) implies \(\dot P\ge n^2/2-C\). Integration gives a uniform bound for \(\int_{t_0+\epsilon}^{t_1}n^2\,\mathop{}\!dt\). 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 \(\mathop{\mathrm{spt}}\nu\) 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 \[ \partial_iU_\alpha=F_{i\alpha} +\partial_jE_{ij\alpha}, \qquad \|F\|\le C,\quad \|E\|=o(1). \tag{156}\] 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 \(C\int e\,\mathop{}\!d\mu=O(a^2)\). 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 (156) 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 6. 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 \[\left\|U-\left(\sum_{X\in\pi^{-1}(y)} \theta(X)z(X)/a\right)\mathop{}\!d\mathop{\mathrm{vol}}_S(y)\right\| =o(1).\] 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 (146) 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 7; points with tilt exceeding a number tending slowly to zero have vanishing mass by (145). 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 \[\sum_{i=1}^Q|w_i|^2 =\sum_{i=1}^Q|w_i-\bar w|^2+Q|\bar w|^2\] 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 \[ t(y)=\sup_{x\in B_5}(\ell_x-|x-y|)_+, \qquad y\in B_2. \tag{159}\] 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 (147), 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 26. Each \(f_i\) is minimal. The mean-curvature operator and its first derivative involve at most three derivatives of the graph. Substitution therefore gives \[|H_{\operatorname{graph}f_\epsilon}|+ |\nabla H_{\operatorname{graph}f_\epsilon}| \le Cm_0^2t_\epsilon^{2k_0-4}.\] 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 (157), 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 27 strictly inside the original failure ball. Small slope and the much smaller height displacement then imply, in the fixed horizontal projection, \[ \lambda(B_\ell(x))\ge c\ell^m \quad\hbox{for every failure }(x,\ell). \tag{160}\] 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 (143). If \(|x|\le K\ell\), then \[c\ell^m\le\lambda(B_\ell(x)) \le\lambda(B_{|x|+\ell}(0)) \le\eta(K+1)^m\ell^m,\] 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 \[\ell\le\frac r{K-A},\qquad B_\ell(x)\subset B_{C_A r}(0),\qquad C_A=1+\frac{A+1}{K-A}.\] Combining (160) with the little-\(o\) part of (143) now gives \[ \sup\{\ell:B_{A\ell}(x)\cap B_r(0)\ne\varnothing, \ (x,\ell)\hbox{ a failure}\}=o(r). \tag{161}\] 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 (161). Select disjoint original balls \(B_i\) whose fivefold enlargements cover their union. The charges in (160) are disjoint, so \[|\mathcal I\cap B_r| \le C\sum_i|B_i| \le C\sum_i\lambda(B_i) \le C\lambda(B_{2r})=o(r^m).\] 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 (140) 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 \[ \ell-|x-y|>t(y)/2. \tag{167}\] 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 \(x\), 27 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 27 once.

For \(y\) with \(\rho(y)<r\), the witnessing scale satisfies \(\ell=o(r)\) by (161). There is a ball of radius \(c_1\ell\) at distance \(O(\ell)\) from \(y\), lying inside \(\rho<r\), on which \[ \inf\phi_r\ge c\sup_C\phi_r, \qquad \inf\psi_r\ge c\sup_C\psi_r, \qquad \inf\psi_r\ge c(\ell/r)^{p-1}. \tag{168}\] 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 (168); 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 27. 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 \[ \delta(r)r\ge c m_0\ell^{k_0}. \tag{169}\] 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 (167); (157) and (169) give \[c m_0\ell^{k_0}\le\delta(r)r \le C m_0\ell^{k_0}.\] The inward ball has mass at least \(c\ell^m\), height at least \(cm_0\ell^{k_0}\), and \(h_r\ge3/4\). Consequently \[\frac{H(r)}{r^2} \ge c\delta(r)^2(\ell/r)^{m+p-1} \ge c(m_0)\delta(r)^{2+(m+p-1)/k_0}.\] 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 (164) follows from bounded normalized mass and \(h_r\le1\). The bound \(\delta(r)\le Cr\) follows from 28. This proves (164).

For any point with \(t>0\), its witness and (169) give \[|H_S|+|\nabla H_S| \le Cm_0^2\ell^{2k_0-4} \le Cm_0\ell^{k_0-4}\delta(r)r \le C\delta(r)r.\] At \(t=0\) the left side is zero. This proves (165).

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 (168). Hence \[\int_C\phi_r\bigl(|H_S|^2+|\nabla H_S|^2\bigr)\,\mathop{}\!d\mu \le Cm_0^2\ell^{2k_0-8} \int_{B_C}\phi_r|z|^2\,\mathop{}\!d\mu.\] 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 (166). ◻

Proof. Write the restricted varifold as \(\vartheta\,\mathcal H^m\llcorner S\), allowing the integer density \(\vartheta\) to be zero off its original support. This is the standard constancy argument; compare (Simon 2018, chap. 8, Theorem 4.1). Tangential first variation gives \(\int_S\vartheta\,\mathop{\mathrm{div}}_S X=0\) for all compactly supported tangent fields. Thus \(\vartheta\) is constant almost everywhere on a connected smaller disk. The constant is \(Q\), by (140). 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 [lt:scalar,sx:signed] 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 7 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 \[|D^j\vartheta_i|\leq C_j\ell^{-j}\qquad(0\leq j\leq k),\] where \(D\) denotes base-coordinate derivatives and \(k=k(m)\) is the finite test order in 6. 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 (148) gives \[\int\xi^2e\,\mathop{}\!d\mu \leq C\int\left[ \bigl(|\nabla_S\xi|^2+|\mathrm{II}|^2\xi^2\bigr)|z|^2 +\xi^2|H_S||z|\right]\,\mathop{}\!d\mu.\] 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 \[ \ell^{-m}(D+\Lambda)(B_{4\ell}) \leq C(\delta/\sigma^2)^2 \leq C\delta^{2-4\beta}. \tag{175}\] 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 7 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 (16) used in the proof of 7, with (175), 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 7 bounds \(|M|\) on each smaller ball by the enlarged \(D+\Lambda\) mass. In particular, applying it with fixed interior room and using (175) also on the larger ball gives \[ \begin{split} |M|(B_{4\ell}(x_i)) &\leq C(D+\Lambda)(B_{8\ell}(x_i)),\\ \ell^{-m}\tau(B_{4\ell}(x_i)) &\leq C\delta^{2-4\beta}. \end{split} \tag{176}\] All these balls lie inside \(B_r\) by the choice of \(c_0\). Summation with comparable weights gives \[ r^2\int^{(r)}\phi_{\rm bulk}\,\mathop{}\!d|M| \leq Cr^2\int^{(r)}\phi\,\mathop{}\!d(D+\Lambda) \leq C(E+r^2H), \tag{177}\] using (173) to absorb a small multiple of \(E\). For every member of the unit multiplier class this also gives \(r^2|\int^{(r)}a\phi_{\rm bulk}\,\mathop{}\!dM|\leq C(E+r^2H)\).

For the signed estimate, invoke 13 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 \[d_i=\max\{C\ell^{-m}\tau(B_{4\ell}(x_i)),\ \delta^{2J_0+4}\}, \qquad T_i=-\log d_i,\] with the fixed enlargement and constant sufficient for 13. Equation (176) 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 \[\ell+\sup|z|\leq C\delta^{2\beta} \leq \exp(-\kappa T_i)\] for some fixed \(\kappa>0\) and all small \(r\); choose, for example, \(\kappa(2J_0+4)<2\beta\). Thus both smallness hypotheses of 13 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 \(\mathop{\mathrm{Lip}}w_i\leq1\). Restoring the physical measure factor \(\ell^m\) and then the normalization \(r^{2-m}\) gives \[ \begin{split} 2r^2\int^{(r)}\phi_{\rm bulk}\,\mathop{}\!dM &\leq r^2\int^{(r)}\phi_{\rm bulk}\,\mathop{}\!dD +C_Q r^2\int^{(r)}\phi_{\rm bulk}\,\mathop{}\!d\Lambda\\ &\quad+C r^{2-m}\sum_i\phi_i\ell^m d_i e^{-F_Q(T_i)}. \end{split} \tag{178}\] 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 (176): \(\phi\) is comparable to \(\phi_i\) throughout \(B_{8\ell}(x_i)\), and these balls have bounded overlap inside \(B_r\). Therefore \[\begin{split} \sum_i\phi_i\ell^m d_i &\leq C\sum_i\phi_i\tau(B_{4\ell}(x_i)) +C\delta^{2J_0+4}\sum_i\phi_i\ell^m\\ &\leq C\sum_i\phi_i(D+\Lambda)(B_{8\ell}(x_i)) +C\delta^{2J_0+4}\sum_i\phi_i\ell^m\\ &\leq Cr^m\int^{(r)}\phi\,\mathop{}\!d(D+\Lambda) +Cr^m\delta^{2J_0+4}. \end{split}\] 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 (178) is at most \[Cg\left(r^2\int^{(r)}\phi\,\mathop{}\!d(D+\Lambda) +r^2\delta^{2J_0+4}\right).\] The floor term \(r^2\delta^{2J_0+4}\) is at most \(C\delta^{J_0+4}H\) by (171). The weighted comparison (173) controls the remaining integral by \(C(E+r^2H)\) and bounds the \(C_Q\Lambda\) term in (178) by \(Cr^2H+o(g)E\), since \(r^2\delta^2=o(g)\). This proves the second line of (174) 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 \(j(\rho,\omega)\,\mathop{}\!d\rho\, \mathop{}\!d\omega\). Throughout this thin calculation \(\|a\|_{1,r}\leq1\), and all constants are uniform over that class. Initially put \[F(\rho,\omega)=-j(\rho,\omega)^{-1} \int_\rho^r a(t,\omega)\phi_{\rm thin}(t) j(t,\omega)\,\mathop{}\!dt, \qquad X=F\mathbf u.\] Then \(\mathop{\mathrm{div}}_S X=a\phi_{\rm thin}\). 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 \[\mathop{\mathrm{div}}_S X=a\phi_{\rm thin}+q_c, \qquad |q_c|\leq C\sigma^{p+1}\phi,\] and is supported where \(\sigma r\leq r-\rho\leq2\sigma r\). The same integration, also differentiated in angular directions, gives \[ |X|/r+ |\nabla_SX-a\phi_{\rm thin}\mathbf u\otimes\mathbf u| \leq C\sigma^2\phi. \tag{179}\] 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 (179); 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 4. 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 \(W=\int^{(r)}\phi|z|^2\,\mathop{}\!d\mu\leq C(H+\delta^2E)\). Equation (179) gives \(|X|\leq C\sigma^2r\phi\) and \(|\nabla_SX|\leq C\phi\). After multiplication by \(r^2\), the three error types in 4 are therefore bounded by \[C\delta rE,\qquad C\sigma^2r^3W,\qquad C\sigma^2r^2\sqrt{EW},\] 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 \[ r^2\left|\int^{(r)}a\phi_{\rm thin}\,\mathop{}\!dM\right| \leq Cr^2\int^{(r)}\phi_{\rm thin}B[\mathbf u,\mathbf u]\,\mathop{}\!d\mu +\eta_1 E+Cr^2H. \tag{180}\] Since \(\rho\geq r/2\) on the thin part, \[\phi_{\rm thin}\leq C\sigma^2(\rho/r)^2\psi, \qquad B[\mathbf u,\mathbf u]\leq2b+C|z|^2e.\] Thus the first term of (180) 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 \[r^2\left|\int^{(r)}a\phi\,\mathop{}\!dM\right| \leq C(E+r^2H)+\eta_1R\qquad(\|a\|_{1,r}\leq1).\] Homogeneity now gives the first line of (174) 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 \(\int\phi_{\rm bulk}\,\mathop{}\!dD\leq\int\phi\,\mathop{}\!dD\). Because \(\eta_1=o(g)\) by (171), 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 (148) and (173) give \[ \left|E-\int^{(r)}\rho\psi z\cdot P_{vh}v\,\mathop{}\!d\mu\right| \leq Cr^2H+\eta E. \tag{184}\] 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 \[|P_{vh}v|^2=v\cdot(B-B^2)v =b-|Bv|^2 \leq b-\frac{b^2}{|v|^2}=bh_r.\] Cauchy–Schwarz therefore bounds the absolute value of the flux in (184) by \(\sqrt{RH}\). Division by \(H\) proves (181).

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 \(\nabla_S\phi=-(\rho/r^2)\psi\mathbf u\), \[\nabla(\phi\circ\pi)=-\frac{\rho}{r^2}\psi K_z\mathbf u.\] 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 \[\begin{split} P_T:D\bigl((\phi\circ\pi)\overline Z_0\bigr) &=\phi P_T:D\overline Z_0 -\frac{\rho^2}{r^2}\psi (K_z\mathbf u)\cdot(I-B)(J_z\mathbf u)\\ &=\phi P_T:D\overline Z_0-\frac{\rho^2}{r^2}\psi +\frac{\rho^2}{r^2}\psi B[K_z\mathbf u,J_z\mathbf u]. \end{split}\] 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 4 gives \[\begin{split} P_T:D\overline Z_0 &=\mathop{\mathrm{div}}_SZ_0-B:\nabla_SZ_0+\varepsilon_Z,\\ |\varepsilon_Z| &\leq C\left\{|z|e|\nabla_SZ_0| +\left[(|\nabla^\perp H_S|+|z|)|z|+\sqrt e\,|z|\right]|Z_0|\right\}. \end{split}\] This is the estimate for the unweighted lift, so the product formula multiplies its entire remainder by \(\phi\). Subtract the reference identity \(Q\int_S\mathop{\mathrm{div}}_S(\phi Z_0)\,\mathop{}\!d\mathop{\mathrm{vol}}_S=0\), in which \(\mathop{\mathrm{div}}_S(\phi Z_0)=\phi\mathop{\mathrm{div}}_SZ_0-\rho^2\psi/r^2\). Since \(\dot A=(2-m)A+\int^{(r)}\rho^2\psi\,\mathop{}\!dM\), we obtain, with \[\widetilde R=\int^{(r)}\rho^2\psi B[K_z\mathbf u,J_z\mathbf u]\,\mathop{}\!d\mu,\] the exact identity \[ \begin{split} \dot A={}&2A-E+\widetilde R +r^2\int^{(r)}\phi(\mathop{\mathrm{div}}_SZ_0-m)\,\mathop{}\!dM\\ &-r^2\int^{(r)}\phi B:(\nabla_SZ_0-I)\,\mathop{}\!d\mu +r^2\int^{(r)}\phi\varepsilon_Z\,\mathop{}\!d\mu. \end{split} \tag{185}\]

In geodesic normal coordinates \(y\), Gauss’ lemma gives \(Z_0=y^i\partial_i\). If \(g^S_{ij}\) are the metric components, then \[f:=\mathop{\mathrm{div}}_SZ_0-m=y\cdot D\log\sqrt{\det(g^S_{ij})}.\] The logarithmic volume factor has vanishing first derivative at \(0\) and bounded second derivative in the fixed chart. Consequently \[|f|\leq C\rho^2,\qquad |\nabla_Sf|\leq C\rho, \qquad \|a_r\|_{1,r}\leq K_{\mathrm{geo}},\quad a_r=f/r^2,\] 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 (172) applies. The first line of (174) and positivity of \(B\) now give \[\begin{split} r^2\left|\int^{(r)}\phi(\mathop{\mathrm{div}}_SZ_0-m)\,\mathop{}\!dM\right| &\leq Cr^2(E+r^2H)+r^2\eta R,\\ r^2\left|\int^{(r)}\phi B:(\nabla_SZ_0-I)\,\mathop{}\!d\mu\right| &\leq Cr^2E. \end{split}\] 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 (173), followed by Cauchy–Schwarz, bound its three terms respectively by \[\begin{split} r^2\int^{(r)}\phi|\varepsilon_Z|\,\mathop{}\!d\mu &\leq C\delta rE+Cr^3(H+\delta^2E) +Cr^2\sqrt{E(H+\delta^2E)}\\ &\leq Cr^2(E+H)+\eta E. \end{split}\] 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 \[ Cr\sqrt R\left(\int^{(r)}\psi|z|^2e\,\mathop{}\!d\mu\right)^{1/2}, \tag{186}\] by Cauchy–Schwarz. Since \(e\leq m\), the second line of (173) gives \[|\widetilde R-R| \leq Cr\sqrt{RH}+C\delta r\sqrt{R(E+R)} \leq Cr\sqrt{RH}+\eta(E+R).\] Combining these estimates in (185) and dividing by \(H\) proves (182). The differentiated kernel has been exhausted by \(\widetilde R\); every other remainder in that identity carries the weight \(\phi\).

Finally, set \[Y=\frac{\rho K_z\mathbf u}{|K_z\mathbf u|^2}.\] 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 \[ \dot H=\int^{(r)}\psi\left\{ |z|^2(\mathop{\mathrm{tr}}_T DY-mh_r) +\frac{2\rho z\cdot P_{vh}K_z\mathbf u}{|K_z\mathbf u|^2} \right\}\,\mathop{}\!d\mu. \tag{187}\] 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, \[DY=P_h+O(\rho+|z|/\rho)=P_h+O(r).\] It follows that \(\mathop{\mathrm{tr}}_TDY-mh_r=m(1-h_r)-e+O(r)\). The positive \(m(1-h_r)\) contribution is bounded by \(C\delta^2(E+R)\), by the proof of (173). 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 (187) by one costs at most \(C\delta r\sqrt{RH}\leq CrH+\eta R\). Equation (184) now proves the upper bound in (183).

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 \(r^{-m}\int_{\rho<2\epsilon}|z|^2\,\mathop{}\!d\mu\), which tends to zero. The preceding support bound and local mass bounds also justify all other limiting terms. The lower bound in (183) 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 (174) with \(a=1\), followed by (181), gives \[|P|\leq C(n_*+r^2)+\eta q_* \leq C(\sqrt{q_*}+r^2)+C\eta q_*.\] It follows, for a sufficiently large fixed \(K\) and \(|P|\geq K\), that \[ \frac{q_*}{|P|}\geq c\min\{|P|,\eta^{-1}\}, \qquad v_0\geq cq_*. \tag{189}\] For the first assertion, either \(C\sqrt{q_*}\) or \(C\eta q_*\) is at least a fixed fraction of \(|P|\). The second follows from (182), 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 \[\dot P=v_0-Ph_0\geq cq_*+mP>0.\] 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 (189) and (172) imply \(v_0/P\geq cT^2\). Define \[Z=\log(r^2/A)=T-\log P.\] 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 \[ \dot P\geq-C(rP+gP^2) \qquad\hbox{when }K\leq P\leq T^4. \tag{190}\] Combining (182) and (183) gives \[\dot P\geq 2P-n_*+q_*-2Pn_* -C\{r\sqrt{q_*}+r^2(1+n_*)+rP\} -C\eta(1+P)(n_*+q_*).\] 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 (181) gives \[q_*\geq n_*^2-Cr^2P-C\eta P^2.\] The independent signed second line of (174), with \(\eta_1\leq\eta\), gives \[n_*\geq2P-d_0,\qquad d_0=C(gP+r^2+\eta P^2).\] For large \(K\) and small \(r\), this also gives \(n_*>1\). The principal terms factor as \[2P-n_*+n_*^2-2Pn_*=(n_*-1)(n_*-2P)\geq-CPd_0.\] 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 (190).

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 \[\int_{s_0}^{\infty}C\{e^{-s}+T(s)^{-16}\}\,\mathop{}\!ds \leq\varepsilon_0<1.\] This is possible by (172). On each component where \(P\geq K\), before a possible exit through \(P=T^4\), (190) yields \[(\log P)_s\leq C(r+gP)\leq C(r+T^{-16}).\] Moreover \(Z_s>0\) there by (189), increasing \(K\) if necessary. If the first component starts at \(s_0\), these inequalities give \[P\leq e^{\varepsilon_0}P_0\leq e^{\varepsilon_0}T_0^2, \qquad T=Z+\log P\geq T_0-\log P_0+\log K\geq T_0/2.\] 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 \[-K\leq P(r)\leq e^{\varepsilon_0}\max\{K,P_0\} \qquad(0<r\leq r_0).\]

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 32 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 32 and \(n_*\leq C\sqrt{q_*}+Cr^2\) imply \[ \dot P\geq cq_*-C_{B_0}. \tag{191}\] Integrating over a logarithmic interval of fixed length bounds \(\int q_*\,\mathop{}\!d\log r\). Cauchy–Schwarz then bounds \(\int n_*\,\mathop{}\!d\log r\), and integration of (183) proves both sides of the height comparison in (188). 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 (181) bounds \(n_*\) at those radii, proving the last assertion. ◻

Proof. Take the radii \(r_j\) from 33, and write \(r=r_j\). On \(\rho<3r/4\) the weights \(\phi,\psi\) have positive lower bounds depending only on \(p\). Equations (173), (181), and the boundedness of \(n_*,q_*\) give \[ r^{-m}\int_{\rho<3r/4}|z|^2\,\mathop{}\!d\mu +r^{2-m}(D+\Lambda)(B_{3r/4})\leq CH(r). \tag{192}\] 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 (188) with \(\lambda=1/4\) and using \(\psi\leq2p\), \(h_r\leq1\) at radius \(r/4\) gives \[ r^{-m}\int_{\rho<r/4}|z|^2\,\mathop{}\!d\mu\geq cH(r)>0. \tag{193}\]

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 9, 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 7 with room between \(3r/4\) and \(r/2\); its integer is again \(Q\). Its variation estimate and (192) give, for every enlarged ball \(B_{c\epsilon r}(x)\) of this cover, \[(\epsilon r)^{-m}\tau(B_{c\epsilon r}(x)) \leq C_\epsilon H(r)/r^2\longrightarrow0,\] 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 9 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 (192) 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 \[ r^{-m}\int_{\rho<r/4}|z|^{p_*}\,\mathop{}\!d\mu \leq CH(r)^{p_*/2}. \tag{194}\]

Let \(\mathcal I=\{t>0\}\) on the center. By 28, 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 7 gives \[ \begin{split} r^{-m}\pi_\#\mu(\mathcal I\cap B_{r/4}) &\leq Qr^{-m}\mathop{\mathrm{vol}}_S(\mathcal I\cap B_{r/4}) +r^{-m}|M|(B_{r/4})\\ &\leq o(1)+CH(r)/r^2=o(1). \end{split} \tag{195}\] Hölder’s inequality with (194) and (195) now gives \[r^{-m}\int_{\rho<r/4}|z|^2\,\mathop{}\!d\mu \leq CH(r) \left[r^{-m}\pi_\#\mu(\mathcal I\cap B_{r/4})\right]^{1-2/p_*} =o(H(r)),\] contradicting (193). Thus the varifold lies on the center near the origin, and 30 gives regularity. ◻

Completion of the proof of 1. The weighted rectifiable tangent and density theorem (Simon 2018, chap. 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 \[\frac{\mu(\{y:\theta(y)\ne Q\}\cap \mathbf B_r(x))}{r^m} \longrightarrow0.\] 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. 34 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 \(N\subset\mathop{\mathrm{spt}}\mu\) 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 3 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 \[\mathcal H^m_{\epsilon}(N) \leq C_m\sum_i r_i^m \leq C_m\sum_i\mu(\mathbf B_{r_i}(x_i)) \leq C_m\mu(O).\] Let \(\mu(O)\downarrow0\) and then \(\epsilon\downarrow0\). An exhaustion of \(U\) proves \(\mathcal H^m\ll\mu\) on \(\mathop{\mathrm{spt}}\mu\). Applying this to the singular set proves \(\mathcal H^m(\operatorname{Sing}(V))=0\). ◻