Proof. If \(t(w)>0\) and \(\rho(w)<r\), choose a failure \((x,\ell)\) for which \(\ell-|w-x|>t(w)/2\). Since \(|w|\le r\), either \(|x|\le16\ell\) or \(16\ell<r+\ell\). Thus \[ \ell\le\max(b,r/15)\le r/15. \tag{26}\] Equation (20) gives \(\delta\le Cr^{k-1}\). The projected cylinder is contained in an ambient ball of comparable radius, so monotonicity gives its normalized mass bound.

If \(\delta=0\), the support lies on \(S\) in a neighborhood of ambient \(0\): its projection lies inside \(\{\rho<r\}\) because its distance from the radial center is \(O(m_0b^k)\). Tangential stationarity then makes the multiplicity, extended by zero on the disk, constant. It is positive because \(0\) belongs to the support. Normal stationarity and elliptic regularity make \(S\) smooth and minimal there. This contradicts the singularity of \(0\).

We next locate the height that pays for a witness. The argument is the weighted construction of (OpenAI 2026, Proposition 7.4), with (26) replacing its typical-point estimate. If the witness is near the kernel boundary, move inward by \(\ell\) along the radial geodesic and take a ball of radius a sufficiently small fixed multiple of \(\ell\); otherwise take a nearby such ball. This ball lies within the kernel support, and its center remains within \(B_{30\ell}(x)\) in the fitting coordinates. On it \(\phi\) and \(\psi\) dominate, up to fixed constants, their suprema on a small cell containing the original point, and \(\psi\ge c(\ell/r)^{p-1}\). Lemma 5 supplies separated entries above a set of base volume \(c\ell^m\). Two entries separated in a nearly vertical direction cannot both lie within a sufficiently small multiple of their separation from a small-slope graph. After transferring to \(S\), at least one entry therefore has height at least \(cm_0\ell^k\) and \(h\ge3/4\). Small slopes and the height bounds make changes of horizontal argument smaller than the available subball margin. The projection Jacobian bound converts base volume into the same lower bound for mass.

In particular \(\delta r\ge cm_0\ell^k\). At a point approaching the height supremum, the witness also satisfies \(\delta r\le Cm_0\ell^k\), by (20). Hence \[H/r^2\ge c\delta^2(\ell/r)^{m+p-1}, \qquad \ell/r\ge c\delta^{1/k},\] where the last inequality uses \(r\le1\). This proves the lower bound in (23), for example with \(J_0\ge2+(m+p-1)/k\); its upper bound follows from the mass bound. The comparison \(\delta r\ge cm_0\ell^k\), together with \(k>4\), proves (24).

To prove (25), group witnesses by dyadic scale and use each covering grid cell once at its scale. By (20), the curvature-square integral on a cell of size \(\ell\) is at most \(Cm_0^4\ell^{4k-8+m}\sup\phi\). Its paying ball contributes at least \(cm_0^2\ell^{2k+m}\sup\phi\) to the height-square integral. At each scale the paying balls have bounded overlap, since they are moved by only \(O(\ell)\). Summing the factors \(Cm_0^2\ell^{2k-8}\) proves the assertion. The cells may be used as an overcover, so no measurable choice of witnesses is necessary.

Finally choose a smooth decreasing radial mass test on a small fixed fraction of \(r\). Compare it with the corresponding ambient radial test at \(0\). The displacement of the center is \(O(m_0b^k)\), the support height is \(O(r^k)\), and chord and geodesic radii on \(S\) differ by \(O(r^3)\). The radius replacement has error \(O(r^2)\), so the normalized averages differ by \(O(r)\), including their reference normalizations. All support encountered by the ambient test lies in the cylinder and has projected radius below \(r/2\). The ambient average is close to \(Q\) by the density-ratio requirement at the start. Decreasing the fixed parameters gives the stated \(1/8\) bound. ◻

Proof. We verify the transfer of the calculation in (OpenAI 2026, Lemmas 8.1–8.2); this also identifies the precise single-radius inputs. The definitions and positivity of \(B\) give \[ \begin{aligned} r^{-m}\int\phi|z|^2\,\mathrm d\mu&\le C(H+\delta^2E_*),\\ r^{-m}\int\psi|z|^2\,\mathrm d\mu &\le C\{H+\delta^2(E_*+R_*)\},\\ r^{-m}\int\phi\,\mathrm d\Lambda&\le C(H+\delta^2E_*). \end{aligned} \tag{32}\] For the first bound use \(1-h\le Ce\); for the second split at \(r/2\) and use \(1-h\le Cb_z\) on the outer part. The last bound also uses (25), with physical units equal to the current start units.

Here are the details of the mass comparison, where preserving the coefficient one in (28) is essential. For the fixed \(0<\beta_1<1/4\) put \(\gamma_1=\delta^{\beta_1}\). Away from the outer collar we use signed excess; on the collar we use a radial first-variation test. Choose a radial cutoff \(\chi\) equal to one at relative depth \(1-\rho/r\le\gamma_1^2\) and zero at depth at least \(2\gamma_1^2\), and put \[\phi_{\rm thin}=\chi\phi,\qquad \phi_{\rm bulk}=(1-\chi)\phi.\] Cover the support of \(\phi_{\rm bulk}\) by cells centered at \(x_j\) of radius \(\lambda_1=c_1\gamma_1^2r\), with a smooth partition of unity \(\vartheta_j\) and bounded overlap of fixed enlargements. Choose \(c_1\) small enough that these enlargements stay inside \(\{\rho<r\}\) and \(\phi\) is comparable there to \(\phi_j=\phi(x_j)\). Their mass is \(O(\lambda_1^m)\) because \(\delta r=o(\lambda_1)\). Testing normal stationarity with a cutoff times \(z\) and using (24) yields \[\lambda_1^{-m}(D+\Lambda)(B_{c\lambda_1}) \le C\delta^{2-4\beta_1}.\] This is the local estimate in (OpenAI 2026, Equation (8.6)). The scalar estimate selects an integer on each cell; the small exceptional projection sets cannot fill the prescribed overlaps, so the integers agree throughout the connected bulk; call their value \(q\). Put \(\epsilon=C\delta^{2-4\beta_1}\). Partition the smooth mass test from Lemma 6 over the cells. Each piece has the scaled derivative norm in Equation (11) bounded by \(Cr^{-m}\). That homogeneous estimate bounds its error relative to \(q\) by \(C\epsilon\lambda_1^m r^{-m}\), without dividing by the piece’s possibly small integral. Bounded overlap gives \(\sum\lambda_1^m\le Cr^m\), so the total error is \(O(\epsilon)\). Since the test average is within \(1/8\) of \(Q\), integrality gives \(q=Q\).

The scalar estimate now bounds \(|M|(B_{4\lambda_1}(x_j))\) by \(C(D+\Lambda)(B_{8\lambda_1}(x_j))\). Summing with comparable weights gives the absolute bulk estimate \[ r^{2-m}\int\phi_{\rm bulk}\,\mathrm d|M| \le Cr^{2-m}\int\phi\,\mathrm d(D+\Lambda) \le C(E_*+r^2H). \tag{33}\] In the last inequality a small multiple of \(E_*\) is absorbed using (32).

For signed excess, use fresh charts on the cells and set \[d_j=\max\{C\lambda_1^{-m}\tau(B_{4\lambda_1}(x_j)), \delta^{2J_0+4}\}, \qquad T_j=-\log d_j,\qquad \tau=|M|+D+\Lambda.\] The local bounds and the floor give \(c|\log\delta|\le T_j\le(2J_0+4)|\log\delta|\). The physical smallness hypothesis follows from \[\lambda_1+\sup|z|\le C\delta^{2\beta_1}, \qquad \kappa(2J_0+4)<2\beta_1.\] These are precisely the fixed parameter choices in Equation (17); the required derivative orders were chosen after \(J_0\) and \(\kappa\), and before the fitting exponent. The partitioned tests \(\vartheta_j\phi_{\rm bulk}/(C\phi_j)\) have size and Lipschitz constant at most one in cell units. Applying signed excess to these tests and restoring the measure factors gives \[\begin{align*} 2r^{2-m}\int\phi_{\rm bulk}\,\mathrm dM &\le r^{2-m}\int\phi_{\rm bulk}\,\mathrm dD +C_Qr^{2-m}\int\phi_{\rm bulk}\,\mathrm d\Lambda\\ &\quad+Cr^{2-m}\sum_j\phi_j\lambda_1^m d_j e^{-F_Q(T_j)}. \end{align*}\] The weighted budgets satisfy \[\sum_j\phi_j\lambda_1^m d_j \le C\int\phi\,\mathrm d(D+\Lambda)+Cr^m\delta^{2J_0+4},\] by scalar control on the enlarged cells and bounded overlap. The exponential factor is smaller than any fixed negative power of \(T\), and the floor costs \(r^2\delta^{2J_0+4}\le C\delta^{J_0+4}H\) by (23). The signed-excess remainder is therefore at most \(Cg(E_*+H)\). The separate \(C_Q\Lambda\) term is bounded by \(Cr^2H+Cr^2\delta^2E_*\); since \(r^2\delta^2=o(g)\), it contributes only \(Cr^2H+CgE_*\). This proves the bulk signed comparison with leading tilt coefficient one.

For the thin part, let \(a\) be a multiplier with \(\sup(|a|+r|\nabla_Sa|)\le1\). The radial primitive constructed in (OpenAI 2026, Lemma 8.1, Equation (8.11)), cut off at relative depths between \(\gamma_1\) and \(2\gamma_1\), gives a compactly supported tangent field \(X\) with \[\operatorname{div}_SX=a\phi_{\rm thin}+q_c,\qquad |X|/r+|\nabla^SX-a\phi_{\rm thin}u\otimes u| \le C\gamma_1^2\phi.\] Here \(q_c\) is supported on that thicker cutoff collar and \(|q_c|\le C\gamma_1^{p+1}\phi\). Its mass contribution is controlled by the absolute bulk estimate (33). In horizontal stationarity, the principal derivative of \(X\) pairs with radial tilt \(B[u,u]\); the remaining derivative has the small factor \(\gamma_1^2\). The resulting bound is \[\begin{aligned} r^{2-m}\left|\int a\phi_{\rm thin}\,\mathrm dM\right| &\le Cr^{2-m}\int\phi_{\rm thin}B[u,u]\,\mathrm d\mu +C\delta^\alpha E_*+Cr^2H \\ &\le C\delta^{\alpha'}(E_*+R_*)+Cr^2H. \end{aligned}\] For the last inequality use, on the thin collar, \(\phi_{\rm thin}\le C\gamma_1^2(\rho/r)^2\psi\) and \(B[u,u]\le2b_z+C|z|^2e\). The errors use only horizontal stationarity, fixed geodesic geometry, and (32). Combining the absolute bulk and thin estimates proves (27), including its version with any bounded scaled \(C^1\) multiplier. For the signed estimate take \(a=1\) and combine with the signed bulk bound. The small tilt error is absorbed in \(CgE_*\), since every positive power of \(\delta\) is \(o(g)\); the radial error remains \(\eta R_*\). This proves (28).

Normal stationarity gives the flux estimate \[\left|E_*-r^{-m}\int\rho\psi\, z\cdot(P_T)_{vh}v_z\,\mathrm d\mu\right| \le Cr^2H+\eta E_*.\] Projection algebra gives the exact inequality \(|(P_T)_{vh}v_z|^2\le b_zh\); Cauchy–Schwarz proves (29). Horizontal stationarity with the lift of \(\rho u\) gives (30), as in (OpenAI 2026, Lemma 8.2). Its geodesic volume-distortion multiplier has bounded scaled \(C^1\) norm, so the preceding absolute estimate applies with a uniform constant.

One modification is needed for the height identity because our center need not pass through ambient \(0\). Off the central fiber put \[Y=\frac{\rho K_zu}{|K_zu|^2}.\] Then \(DY=P_h+O(\rho+|z|)\), where \(P_h\) is horizontal projection at the nearest point of \(S\). To see this, the pulled-back radial field \(\rho u\) has derivative \(P_h+O(\rho+|z|)\). Its correction in \(Y\) is \(\rho\) times a function of the tube coordinates and \(u\) that is \(O(|z|)\), including its angular derivatives. Angular differentiation therefore costs \(\rho\cdot O(|z|)/\rho=O(|z|)\), and the remaining derivative costs \(O(\rho)\). Testing with \(\psi|z|^2Y\) gives the following identity, whose integrand is extended by zero on the central fiber: \[ \dot H=r^{-m}\int\psi\left\{ |z|^2(\operatorname{tr}(P_TDY)-mh) +\frac{2\rho z\cdot(P_T)_{vh}K_zu}{|K_zu|^2} \right\}\,\mathrm d\mu . \tag{34}\] This is (OpenAI 2026, Equation (8.18)). First omit the central fiber using a radial cutoff. The transition term tends to zero because \(Y=O(\rho)\) and the mass of the open shrinking transition annulus tends to zero. On the fiber \(h=0\) almost everywhere, so the left-hand side is unchanged; the flux term has a factor \(\rho\). The error in \(DY\) is \(O(r)\) on the support, and (32) and the normal flux estimate give the upper bound in (31). The lower bound follows by direct differentiation of the nonnegative weight \(r^{-m}\psi\). All parameter choices have been fixed independently of the start. ◻

Proof. Equations (27) and (29) imply \(|P|\le C(\sqrt{q_*}+r^2)+C\eta q_*\). For \(|P|\) large, either the square-root term or the last term controls a fixed fraction of \(|P|\), proving the first assertion of (35). Equation (30) then gives its second assertion, after increasing \(K\) and decreasing \(r_*\).

For positive \(P\), quotient differentiation and the upper height bound give the principal expression \(2P-n_*+q_*-2Pn_*\), with the errors displayed in Lemma 7. If \(q_*\) is larger than a sufficiently large fixed multiple of \(P^2\), it absorbs all negative terms, since \(n_*\le C\sqrt{q_*}+Cr^2\) and \(\eta T^4=o(1)\). Otherwise \(n_*\le CP\) and \[q_*\ge n_*^2-Cr^2P-C\eta P^2, \qquad n_*\ge2P-C(gP+r^2+\eta P^2).\] For large \(K\) the principal terms factor as \((n_*-1)(n_*-2P)\), bounded below by \(-CP(gP+r^2+\eta P^2)\). Since \(\eta T^4\le g\) at small radii, the term \(\eta P^3\) is absorbed by \(gP^2\). This proves (36).

If \(P\le-K\), the lower height bound gives \(\dot P=v_0-P\dot H/H\ge cq_*-m|P|>0\). Thus \(P\) decreases in inward time \(-\log r\) whenever it is below \(-K\). This proves the finite-interval lower bound from the inner endpoint. On an infinite interval, such an excursion would also make \(A<0\) decrease inward and remain below a fixed negative value. That is impossible because \(A\to0\): its normalized mass integral is bounded and it has a factor \(r^2\).

To prove the upper bound, work on a component where \(P\ge K\) before any exit through \(P=T^4\). In inward time, (36) gives \[(\log P)'\le C(r+T^{-16}).\] The integral of this right-hand side from a sufficiently small radius to zero is less than one, since \(T\ge2|\log r|-C\). Moreover \(Z=T-\log P=\log(r^2/A)\) satisfies \(Z'=v_0/P-2>0\) by (35). If the component starts at \(r_0\) with \(P_0\le T_0^2\), then \(P\le\exp(1)P_0\) and \(T\ge T_0-\log P_0+\log K\ge T_0/2\). For large \(T_0\) these inequalities exclude \(P=T^4\). Components starting later at \(P=K\) satisfy the same argument, giving (37).

On an infinite interval, suppose no sufficiently small radius satisfied \(P\le T^2\). Then eventually \(P>T^2\), and (35) implies \(Z'\ge cT^2-2\). The lower bound for \(T\) makes \(Z\) positive eventually. Since \(Z\le T\) in this regime, one then has \(Z'\ge c'Z^2\), which forces blow-up in finite inward time. The continuous positive quantities \(A,H\) are defined at every finite time in this regime, a contradiction.

Finally, for \(|P|\le C_0\), use the upper height derivative bound if \(P\ge0\) and the lower bound if \(P<0\). Absorbing \(\eta q_*\) gives \[\dot P\ge cq_*-C_{C_0}.\] Integration bounds \(\int q_*\,\mathrm d\log r\) on every bounded-ratio interval. Equation (29) and Cauchy–Schwarz bound \(\int n_*\,\mathrm d\log r\), and (31) gives both height comparisons in (38). ◻

Proof. Coverage of the targets. The top of the interval belonging to \(R\) is \((r_*/N_*)R\). Successive tops decrease to zero unless the construction stops, since the next start is at most \(FbR<R/4\). Suppose a target \(L_i\) lies below one top and above the next, but misses the first interval. It must then lie below its lower endpoint, so \(L_i<B_*bR\) and \(FbR>10N_*L_i/r_*\). A dyadic scale between \(2N_*L_i/r_*\) and \(4N_*L_i/r_*\) is therefore eligible for the next start. For all sufficiently late \(i\) it is good, by convergence at \(L_i\) to a plane. Maximality of the next start would make its top larger than \(L_i\), a contradiction. Hence all sufficiently late targets are covered.

A fixed interval containing targets tending to zero necessarily has \(b=0\). On that interval Lemma 8 first supplies a lower bound and a sufficiently small radius with \(P\le T^2\), then an upper bound and height doubling inward. This proves the assertion in the fixed-interval case. After taking a subsequence, we may therefore restrict to targets whose covering interval indices tend to infinity. There are then infinitely many starts, all with positive failure parameter \(b\).

Geometry of the late starts. We first show that the \(b\) values of these covering intervals tend to zero. Otherwise their start scales would be comparable to their target scales, since \(B_*b\le L_i/R\le r_*/N_*\). The start varifolds would then converge to planes, contradicting failed-fit separation at a scale bounded away from zero.

In fact the start varifolds converge to multiplicity-\(Q\) planes even without comparability to the targets. Any subsequential limit is a stationary tangent cone at \(0\), of density \(Q\). Pass also to limits of the start coordinates and their uniformly controlled center graphs. Because \(t(w)\le b+|w|\) and \(b\to0\), the limiting cone lies within \(C|w|^k\) of a \(C^1\) graph through \(0\). Every conical ray is therefore in the tangent plane of that graph at \(0\). Constancy gives a multiplicity-\(Q\) plane. Thus, if a minus sign denotes the preceding interval, eventually \[ R=Fb_-R_-. \tag{42}\] Indeed the right-hand side is dyadic. If it exceeded the chosen \(R\), then \(2R\) would still be eligible; planar convergence of \(V_R\) on every fixed ball makes \(2R\) good, contradicting maximality. It follows also that \(b_-\to0\). If not, the preceding starts would be comparable to the current planar starts, again contradicting separation.

The outer endpoint: the preceding failure. We now work entirely in the current start’s units. The preceding failure will give an upper support-height bound and a matching lower height integral on fixed room. Together these bounds will control \(P(r_*)\). Define the preceding failure height in current units by \[h_0=\frac{m_0b_-^k}{Fb_-}.\] In every required fixed bounded region of the current start, all support points are within \(Ch_0\) of the preceding center, expressed in current units: in preceding units the region has radius \(O(Fb_-)\), and (20) applies with \(t\le b_-+|w|\). The preceding and current start planes make a small angle. To verify this, take a subsequence: the preceding centers in current units have small slope and pass within \(Ch_0\) of \(0\), and the current planar support limit lies in their limiting graphs. The current chosen planes converge to this support plane. This proves the asserted angle bound, allowing a fixed enlargement of the small slope threshold.

The current center consequently satisfies \[ |z|\le Ch_0 \tag{43}\] on the region containing \(\{\rho<4r_*\}\). Indeed a current witness of scale \(\ell\) supplies two entries separated by \(cm_0\ell^k\) in a direction nearly vertical also for the preceding plane. Both lie within \(Ch_0\) of its small-slope center, so \(cm_0\ell^k\le Ch_0\). Equation (20) now proves (43). By Lemma 6 at radius \(4r_*\), the mean curvature and its first normal derivative are at most \(Ch_0\) there as well. Every comparison is internal to both cylinders: the current cylinder has fixed room, whereas its required region in preceding units shrinks with \(Fb_-\to0\).

A preceding failure attaining \(b_-\) supplies the complementary bound \[ H(r_*)\ge ch_0^2. \tag{44}\] Its center is within \(16b_-\) of horizontal \(0\), so the failed-fit lemma may be used on a subball centered near horizontal \(0\). The entries lie within ambient radius \(Cb_-\) in preceding units: their heights over the fit are \(O(m_0b_-^k)\), and the glued center has the same bound at \(0\). In current units they are well inside \(\{\rho<r_*\}\) by (39); their mass is bounded below by a constant times \(F^{-m}\). Their tilt remains small relative to the current center. At least one entry of each separated pair has height at least \(ch_0\) over that center, proving (44).

The cutoff normal test with \(z\) on fixed room around \(r_*\), using (43) and its curvature bound, gives \(E_*(r_*)+R_*(r_*)\le Ch_0^2\). Hence (27) and (44) bound \(|P(r_*)|\) uniformly, while \(T(r_*)\to\infty\). In particular \(P(r_*)\le T(r_*)^2\) for all sufficiently late covering intervals, as required for the inward upper bound.

The inner endpoint: the current failure. For \(b>0\) we use the same height-versus-tilt comparison at radius \(B_*b\), now supplied by the current failure. On a fixed enlargement, \(|z|\le Cm_0b^k\) follows from \(t\le b+|w|\); the curvature bounds of Lemma 6 apply there. The cutoff tilt estimate on room of scale \(b\) gives \[E_*(B_*b)+R_*(B_*b)\le Cm_0^2b^{2k}.\] The inverse-square scale in that estimate is canceled by the factors in (22). A failure attaining \(b\), tested on a subball of radius \(10^{-2}b\) near horizontal \(0\), supplies low-tilt entries well inside the kernel and proves \(H(B_*b)\ge cm_0^2b^{2k}\). Thus \(|P(B_*b)|\) is bounded as well.

The outer bound controls \(P\) from above as the radius decreases. The inner bound controls it from below: once \(P<-K\), it decreases inward, so a value below the inner-endpoint bound could not recover before that endpoint. Lemma 8 therefore gives (41). Its final smallness requirement on \(\eta\) is uniform on late intervals: use \(\delta\le Cr\) on the sufficiently small radii and (43) on the remaining radii. This proves the uniform height comparison. ◻

Proof. Suppose that \(\nu\) vanishes on a nonempty open subset. Because \(\nu\ne0\) and \(\Omega\) is connected, choose a point \(x\) outside \(\operatorname{spt}\nu\) so close to its boundary that \[l_0=\operatorname{dist}(x,\operatorname{spt}\nu)>0, \qquad \overline B_{2l_0}(x)\subset\Omega.\] Then \(H(l)=0\) for \(l\le l_0\) and \(H(l)>0\) for \(l_0<l\le l_1\), where \(l_1>l_0\) is fixed and interior. Put \(t=\log l\), \(t_j=\log l_j\), and on \((t_0,t_1]\) define \[P=A/H,\qquad n=D/H,\qquad q=R/H.\] By (82), \[ n\ge0,\quad n\ge2P,\quad q\ge n^2,\qquad \dot P=2P-n+q-2Pn \ge(n-1)(n-2P). \tag{83}\] For \(P\ge1\) the right-hand side is nonnegative. Therefore, going inward from \(t_1\), \(P(t)\le\max\{1,P(t_1)\}\).

On \(B_{l_0}(x)\), the distributional identities give \(f=0\) and \(d_0=0\), and (79) gives \(\mathfrak m=0\). The compact kernels converge uniformly and vanish on the boundary sphere, so \(A(t)\to0\) as \(t\downarrow t_0\), even if \(\mathfrak m\) charges that sphere. Moreover, \[(e^{-2t}A)^{\displaystyle\cdot} =e^{-2t}(R-D)\ge-\tfrac14e^{-2t}H,\] because \(q-n\ge n^2-n\ge-1/4\). Since \(\dot H=2D\ge0\), integration from \(t_0\) yields \[P(t)\ge-\frac{e^{2(t-t_0)}-1}{8}.\] Thus \(P\) is bounded on this finite interval. Using \(q\ge n^2\) in the exact equation for \(\dot P\) now gives \[\dot P\ge\tfrac12 n^2-C.\] Integrating from \(t_0+\varepsilon\) to \(t_1\) bounds \(\int_{t_0+\varepsilon}^{t_1}n^2\,\mathrm dt\) independently of \(\varepsilon\). In particular \(n\) is integrable down to \(t_0\). But \((\log H)^{\displaystyle\cdot}=2n\), which prevents \(H\) from tending to zero there. This contradiction excludes every nonempty open zero-height region. Since the kernel is positive in its open ball, the stated conclusion follows. ◻

Proof of Proposition 13. The preceding lemma makes \(P(x,l)=A(x,l)/H(x,l)\) well defined at every interior ball. In the harmonic-map model from Section 6, this quotient is half the weighted Dirichlet frequency \[\frac{l^2\int\phi_{x,l}|Du|^2\,\mathrm dy} {\int\psi_{x,l}|u|^2\,\mathrm dy}.\] For a nonzero harmonic map homogeneous of degree \(d\) about \(x\), the flux identity in (82) makes this frequency equal to \(d\), so \(P=d/2\). For the measure data, constant frequency at the retained centers, together with the lower bound \(2P\ge1\) supplied by their density comparison, will force the radial measure identity used below.

At a center in \(Z\), the additional hypothesis (80) gives \(2P\ge1\) for small radii. Hence (83) improves to \[ 2P\ge1,\qquad n\ge2P,\qquad \dot P\ge(n-1)(n-2P)\ge0. \tag{84}\] In particular the finite limit \(P(x,0+)\) exists and is at least \(1/2\).

Suppose that \(\dim_{\mathcal H}Z>m-1\), and choose \(m-1<s'<\dim_{\mathcal H}Z\). We may first restrict to centers in a compact interior region for which (80) holds on one common interval \(0<l<l_*\). To justify this restriction, exhaust \(\Omega\) by compact sets with positive boundary distance and impose the comparison for all rational radii below \(1/k\). The resulting sets are Borel by continuity of the kernels, their union is \(Z\), and continuity in radius supplies the comparison at every radius. One of them still has dimension greater than \(s'\).

The compact-subset theorem and Egoroff’s theorem now give a compact set \(K\) of positive finite \(\mathcal H^{s'}\) measure on which \[ P(y,l)\longrightarrow P(y,0+)\quad\hbox{uniformly as }l\downarrow0. \tag{85}\] Indeed first apply Egoroff at dyadic radii; monotonicity supplies the same conclusion between them. Since each function \(P(\cdot,l)\) is continuous, its uniform limit is continuous on \(K\). A further compact restriction gives \[ \mathcal H^{s'}(K\cap B_u(y))\le C u^{s'}. \tag{86}\] Choose \(x\in K\) with positive upper \(s'\)-density, and radii \(l_j\downarrow0\) such that \(\mathcal H^{s'}(K\cap B_{l_j}(x))\ge c l_j^{s'}\).

We next normalize the measure tuple at these scales. Fix an open neighborhood \(\Omega'\Subset\Omega\) of \(x\), so that the constant in (79) is fixed on \(\Omega'\). Put \(T_j(y)=(y-x)/l_j\) and \(N_j=l_j^m H(x,l_j)\). On the expanding domains \(T_j(\Omega)\) define \[ \nu_j=N_j^{-1}(T_j)_\#\nu,\qquad f_j=l_jN_j^{-1}(T_j)_\# f,\qquad (\beta_j,d_j,\mathfrak m_j) =l_j^2N_j^{-1}(T_j)_\#(\beta,d_0,\mathfrak m). \tag{87}\] The identities and inequalities (77)–(79) are invariant under this normalization, and the new height at center zero and radius one equals one.

Here are the local mass bounds needed to pass to a limit. At the fixed center \(x\), \(P\) is bounded on \(0<l<l_*/2\). The exact quotient equation and \(n\le\sqrt q\) therefore give \[\dot P=q-(1+2P)n+2P\ge\tfrac12q-C.\] On every interval of radii with bounded ratio, this bounds \(\int q\,\mathrm d\log l\). Since \(\dot H/H=2n\le2\sqrt q\), Cauchy–Schwarz gives two-sided comparison of \(H\) on the same interval.

To see the resulting mass estimates explicitly, fix \(R>0\) and discard finitely many \(j\) so that all balls used below lie in \(\Omega'\). Comparison on the interval from \(\min\{1,R\}l_j\) to \(\max\{1,4R\}l_j\) gives \(H(x,\tau l_j)\le C_RH(x,l_j)\) for \(R\le\tau\le4R\). The kernel \(\psi_{x,2Rl_j}\) is at least \(c_Rl_j^{-m}\) on \(B_{Rl_j}(x)\); hence \[\nu(B_{Rl_j}(x))\le C_Rl_j^mH(x,l_j).\] The integrated bound for \(q\) on \([2Rl_j,4Rl_j]\) supplies \(\tau_j\in[2R,4R]\) with \(n(x,\tau_jl_j)\le C_R\). The kernel \(\phi_{x,\tau_jl_j}\) is also at least \(c_Rl_j^{-m}\) on \(B_{Rl_j}(x)\), so the definition of \(D=nH\) gives \[d_0(B_{Rl_j}(x))\le C_Rl_j^{m-2}H(x,l_j),\qquad \nu_j(B_R)+d_j(B_R)\le C_R.\] Positivity bounds \(\beta_j\) by its trace \(d_j\), and Cauchy–Schwarz for (77) bounds \(f_j\). Finally, \(|\mathfrak m_j|\le C_{\Omega'}d_j\) on every fixed ball for all sufficiently large \(j\). In particular the same constant is available on all fixed balls of the eventual limit.

Pass to a diagonal subsequence with local weak limits on \(\mathbb R^m\), and denote them by \[\nu_\infty,\quad f_\infty,\quad\beta_\infty,\quad d_\infty,\quad\mathfrak m_\infty.\] They satisfy the same measure identities and inequalities. The normalization gives \(H_\infty(0,1)=1\), so \(\nu_\infty\ne0\) and Lemma 14 applies on all of \(\mathbb R^m\).

The rescaled center measures \[\lambda_j=l_j^{-s'}(T_j)_\# \big(\mathcal H^{s'}\!\llcorner(K\cap\overline B_{l_j}(x))\big)\] have a nonzero weak limit \(\lambda\) on \(\overline B_1\), with the upper ball bound (86). Consequently \(\operatorname{spt}\lambda\) spans \(\mathbb R^m\). Otherwise its bounded support lies in an \((m-1)\)-dimensional linear space and can be covered by \(O(\varepsilon^{-(m-1)})\) balls of radius \(\varepsilon\). The ball bound would give \(\lambda(\mathbb R^m)\le C\varepsilon^{s'-m+1}\to0\), contradicting its nonzero mass.

We have now retained enough centers in the second blow-up. Its remaining feature is that they all have the same constant frequency. If \(z\in\operatorname{spt}\lambda\), choose \(y_j\in K\) with \(T_j(y_j)\to z\). At every fixed radius \(r>0\), \[P_j(T_j(y_j),r)=P(y_j,l_jr) \longrightarrow P(x,0+)=:P_0.\] This follows from (85) and continuity of the limiting frequency on \(K\). Local mass bounds and continuous compact kernels permit passage to the weak limits even with these moving centers. Since the limiting height is positive, \[ P_\infty(z,r)=P_0 \quad(z\in\operatorname{spt}\lambda\cup\{0\},\ r>0), \qquad P_0\ge\tfrac12. \tag{88}\] The comparison inequality also passes to these centers.

Proof of Theorem 1. Suppose that \(\dim_{\mathcal H}\operatorname{Sing}(V)>m-1\), and choose \(m-1<s<\dim_{\mathcal H}\operatorname{Sing}(V)\). Proposition 2 gives the fixed singular point, flat target scales, and center measures satisfying (4). Proposition 10 supplies the centered, nonzero height normalization at these scales. Proposition 11 then gives on the connected base ball \(B_4\subset\Pi\) the nonzero measure tuple satisfying (57), (58), and (59). These are precisely the hypotheses (77)–(79).

Pass to a weak limit \(\sigma\) of the center measures. It is nonzero, is supported in \(\Pi\cap\overline{\mathbf B}_1\), and retains the upper ball bound with exponent \(s\). The mass distribution principle therefore gives \(\dim_{\mathcal H}\operatorname{spt}\sigma\ge s\). Proposition 12 supplies (65) at every point of \(\operatorname{spt}\sigma\), which is (80). Applying Proposition 13 with \(Z=\operatorname{spt}\sigma\) gives \[s\le\dim_{\mathcal H}\operatorname{spt}\sigma\le m-1,\] a contradiction.

For sharpness, take two distinct \(m\)-planes intersecting in an \((m-1)\)-plane, each with multiplicity one. Their sum is stationary by additivity of first variation. Away from the intersection it is locally a single smooth plane, whereas at the intersection its support is not a smooth embedded \(m\)-submanifold. Its singular set therefore has dimension exactly \(m-1\). ◻