Proof. Begin with \(\widetilde h=-a_*t\). Direct differentiation on the round sphere gives \[ |\nabla_{g_0}\widetilde h|_{g_0}^2 =4a_*^2t(1-t),\qquad \nabla_{g_0}^2\widetilde h =2a_*(tg_0-\mathrm dy\cdot\mathrm dy). \tag{5}\] At every point there is a nonzero tangent vector in \(\ker\mathrm dy\), because \(\mathrm dy:T_xB\to\mathbb R^3\) has a four-dimensional domain. A round unit vector \(e\) in this kernel satisfies \(\mathrm d\widetilde h(e)=0\) and \((\nabla_{g_0}^2\widetilde h)(e,e)=2a_*t\). Since the Hessian in (5) has norm at most \(2a_*\) and the squared gradient has norm at most \(a_*^2\), on \(t\ge1/4\) we obtain \[J_{\widetilde h}(e)\ge a_*/2-2a_*^3\ge a_*/4\] after choosing \(a_*^2\le1/8\).

At a point \(x_*\) with \(t=1\), the Hessian has eigenvalues \(2a_*,2a_*,0,0\), and the gradient is zero. On a small neighborhood of \(x_*\) contained in \(\{t>1/2\}\), two Hessian eigenvalues therefore have a uniform positive lower bound and the gradient is arbitrarily small. In round normal coordinates \(\xi\) at \(x_*\), add \(\varepsilon|\xi|^2/2\) times a fixed smooth cutoff equal to one near \(\xi=0\). Choose \(\varepsilon>0\) sufficiently small that this is a \(C^2\)-small perturbation on its support. The resulting function \(h_0\) has positive definite Hessian at \(x_*\).

Here is why the cutoff region retains the required test. Shrinking its support and then the perturbation if necessary, the perturbed Hessian has two eigenvalues at least \(c a_*\), its norm is at most \(C a_*\), and \(|\nabla_{g_0}h_0|\) is sufficiently small that \(C a_*|\nabla_{g_0}h_0|^2<c a_*/2\). The positive two-plane intersects \(\ker\mathrm dh_0\) in dimension at least one. A unit vector in that intersection gives \(J_{h_0}\ge c a_*/2\). This argument also applies at a critical point. Outside the perturbation support the previous \(\ker\mathrm dy\) test applies.

These conclusions persist in a sufficiently small common \(C^1\) neighborhood of \(g_0\). Indeed, the difference of the two Hessians of a fixed function is the connection difference contracted with its differential, and all metric and gradient comparisons use only \(C^1\) bounds. Outside the perturbation support, the condition \(\mathrm dy(e)=0\) still implies \(\mathrm dh_0(e)=0\) independently of the metric. On the support, the positive two-plane and small-gradient estimates just established are uniform under a small perturbation.

Finally, positive definiteness at \(x_*\) holds on a neighborhood of that point. There is a nearby point with nonzero differential: otherwise \(h_0\) would be constant on that neighborhood. Choose a relatively compact neighborhood \(U_0\) of such a point where the Hessian remains positive definite and the gradient stays separated from zero. Choose \(U_0\Subset V\Subset\Omega\subset\{t>1/2\}\) in a fixed coordinate chart, and shrink \(\mathcal G\) once more. All the stated constants can then be chosen uniformly. Notice that \(U_0\) is chosen away from the critical point \(x_*\). ◻

Proof. The pattern will depend on three variables transverse to the parent gradient frozen at its center. Radial wells in these variables have two positive tangential Hessian directions; a small amplitude controls their potentially negative radial contribution to \(J_v\). We first choose the wells and smaller concentric balls that will supply the repeatable gain. We then place this pattern in the parent and verify the Hessian test, including where the parent cutoff varies.

Proof. Choose \(C_{\mathrm p}\) large enough and \(c_{\mathrm p}>0\) small enough for both Lemma 5 and the seed preparation following Lemma 4. Choose the final \(\nu\) sufficiently small for the former lemma and so that a fixed cube \(P_0\) of side \(\nu\) has closure in \(U_0\). All these choices are uniform for the metrics in \(\mathcal G\). Put \(r=\nu^2\).

Start with the prepared family \(\mathcal P_0=\{P_0\}\) for \(h_0\). At generation \(i\ge1\), apply Lemma 5 on every cube of \(\mathcal P_{i-1}\), using parent scale \(S_i=r^{i-1}\), and call the resulting function \(h_i\). The selected child cubes form \(\mathcal P_i\); their sides are \(\nu r^i\) and they are prepared at scale \(r^i\). Every generation consists of finitely many cubes with disjoint interiors. Each perturbation is supported strictly inside its parent, so each \(h_i\) is smooth, including across all cube boundaries. Nothing outside \(P_0\) changes.

Let \[I_i=\sum_{P\in\mathcal P_i}\int_P|\nabla h_i|\,\mathrm dx.\] Equation (7) gives \[ I_i\ge(1+\zeta)I_{i-1},\qquad \int_V\sqrt{1+|\nabla h_i|^2}\,\mathrm dx \ge I_i\ge I_0(1+\zeta)^i. \tag{20}\] The initial mass \(I_0\) has a positive uniform lower bound by Lemma 4 and the fixed volume of \(P_0\).

We give the derivative accounting for all orders with one growth constant. Let \(\beta\ge1\) and \(0<\alpha\le1\) be the gradient comparison factors of Lemma 5. The largest gradient after \(i\) generations is at most a fixed multiple of \(\beta^i\). On \(P_0\) every point has gradient at least \(s_{\min}\alpha^i\), since a point is changed at most once per generation. At generation \(i\), the center slope of each parent is therefore at most \(C\beta^{i-1}\). By (8), disjointness within that generation gives \[ \|D^j(h_i-h_{i-1})\|_{L^\infty} \le C_j\beta^{i-1}r^{i(1-j)}. \tag{21}\] There is no factor counting cubes in this supremum norm estimate. There is also no product of higher pattern derivative constants over generations: the increments are added, and their individual formulas involve frozen slopes and affine matrices. Passing to the fixed finite atlas only adds fixed chain-rule constants and lower-order derivatives, which satisfy the same upper bound. Summation yields \[\begin{align*} \|h_m\|_{C^j} &\le C_j\left(1+\sum_{i=1}^m \beta^{i-1}r^{i(1-j)}\right)\\ &\le C'_j\exp\left( [\log\beta+(j-1)_+|\log r|+1]m\right) \le C'_j e^{C_*(j+1)m}, \tag{22}\end{align*}\] where \((j-1)_+=\max\{j-1,0\}\) and one finite \(C_*\) works for every \(j\ge0\).

The lower Hessian margin has the same exponential control. At a point outside \(P_0\), the seed margin remains unchanged. At a point in \(P_0\), consider the last generation \(i\le m\) that changes the function there. If no such generation occurs, use the seed margin. Otherwise Lemma 5 bounds its retained margin below by \[c\,\frac{s_{\min}\alpha^{i-1}}{r^{i-1}} \ge c s_{\min}\alpha^m.\] Later perturbations vanish near that point, or all their jets there vanish at their support boundaries, so this margin persists. By increasing \(C_*\) if necessary, \[ \max_eJ_{h_m}(e)\ge c e^{-C_*m} \quad\text{on }\{t\ge1/4\}. \tag{23}\] This argument requires no nonzero-gradient assumption outside \(P_0\); critical points there are covered by the seed lemma.

Fix \(\theta>0\) so small that \(C_*\theta\le\delta\), and for each \(k\ge2\) take \[m=\lfloor\theta\log k\rfloor, \qquad \gamma=\theta\log(1+\zeta)>0, \qquad h=h_m+c_k.\] Equations (22) and (23) give the first two estimates in (19). Equation (20) gives the third, since \((1+\zeta)^m\ge(1+\zeta)^{-1}k^\gamma\). Adding bounded constants affects only the zeroth-order bound, and \(h=-a_*t+c_k\) on \(t\le1/2\) because both the seed perturbation and all corrugations lie above that set.

For clarity about subsequent uses, the constants determining \(C_*\), \(\theta\), and \(\gamma\) use only the fixed pattern, the seed, and common \(C^1\) metric bounds. Higher derivatives of an individual input metric may enter constants in a later smooth coordinate change, but cannot alter these exponents. Indeed, for any fixed such coordinate change, an order-\(j\) derivative of the pullback is a finite sum of derivatives of \(h\) of orders at most \(j\) multiplied by fixed smooth coefficients. Its growth is still bounded by \(C_jk^{\delta(j+1)}\). Each finite order may have its own constant and frequency threshold. This establishes the common exponent for arbitrarily high finite accuracy and completes the proof. ◻

Proof. For \(b>0\), the first identity in (24) is equivalent to \[Dv=bUe_1,\qquad De_1=-Uv/b.\] The two prescribed columns are compatible with symmetry precisely when \[ b^2U(e_1,e_1)+U(v,v)=0. \tag{25}\]

First suppose that \(b\ge k^{-12\delta}\), and set \[c_0=\frac{J'}{1+b^2},\qquad U=H-c_0I.\] Then (25) holds. To give the extension of \(D\) explicitly, write its blocks relative to \(\mathbb Re_1\oplus\mathbb Rv\oplus E\), where \(E=\{e_1,v\}^{\perp}\). Denote the corresponding blocks of \(U\) by \(u_{11},u_{12},u_{1E},u_{2E},\ldots\). Take \[D= \begin{pmatrix} -u_{12}/b & b u_{11} & -u_{2E}/b\\ b u_{11} & b u_{12} & b u_{1E}\\ -u_{E2}/b & b u_{E1} & 0 \end{pmatrix}.\] The identity \(u_{22}=-b^2u_{11}\) verifies both required columns. Moreover, \[H-U=c_0I\ge\tfrac12k^{-2\delta}I,\qquad |U|\le Ck^{4\delta},\qquad |D|\le Ck^{16\delta}.\] These imply (24) for large \(k\).

Now suppose that \(0\le b<k^{-12\delta}\). Put \(K=k^{14\delta}\) and take \[U=-K(I-vv^{\mathsf t})+b^2Kvv^{\mathsf t},\qquad D=-bK(e_1v^{\mathsf t}+ve_1^{\mathsf t}).\] The characteristic identity follows directly. These formulas also give \(D=0\) and \(Uv=0\) when \(b=0\), so no division by a small \(b\) is involved in this case.

For the domination inequality, split \(H-U-k^{-3\delta}I\) into \(\mathbb Rv\oplus v^\perp\). Its \(v^\perp\) block is at least \(\tfrac12KI\) for large \(k\). Its scalar \(v\) block is at least \[k^{-2\delta}-k^{-20\delta}-k^{-10\delta}-k^{-3\delta},\] because \[H(v,v)\ge J'-b^2|H|,\qquad b^2|H|\le k^{-20\delta},\qquad b^2K\le k^{-10\delta}.\] The Schur complement subtracts at most \(C|H|^2/K\le Ck^{-6\delta}\). The positive \(k^{-2\delta}\) term dominates every subtraction. The matrix is therefore positive definite for large \(k\). Finally, \(|U|\le Ck^{14\delta}\) and \(|D|\le Ck^{2\delta}\), proving the remaining bound. ◻

Proof. Choose an index \(i\) with \(|q_i|\ge c/2\), write \(q\cdot\partial=q_i\partial_i+D'\), and let \(I_i\) denote integration in \(X_i\) with zero integration constant. On \(\mathcal P_m\), set \[\mathcal K=\frac{D'I_i}{q_i},\qquad R_{q,m}=\frac{I_i}{q_i}\sum_{r=0}^{m}(-\mathcal K)^r.\] Each application of \(\mathcal K\) lowers the degree in the variables other than \(X_i\). Thus \(\mathcal K^{m+1}=0\) on \(\mathcal P_m\). Since \[(q\cdot\partial)\frac{I_i}{q_i}=I+\mathcal K,\] the finite geometric sum gives \((q\cdot\partial)R_{q,m}=I\). The stated coefficient bound follows from the finite formula and the bounds on \(q_i^{-1}\) and \(q\). The four possible choices of \(i\) have the same uniform bound. ◻

Proof. We divide the construction into the selection of jets, finite polynomial equations, and localization estimates.

Proof. The bounds on the first two derivatives of \(h\) imply \[|D_xQ|\le Ck^{4\delta}\] in the fixed chart. A packet centered at \(z\) that meets the ball about \(x\) has \[|z-x|\le Ck^{-1/3-100\delta}+C/k.\] For large \(k\), such a \(z\), and the whole ball, belong to a compactly contained part of \(\Omega\). Normal coordinates centered at \(z\) are defined throughout the ball. The preceding estimates give \[|Q_z-Q_x| \le Ck^{-1/3-96\delta}+Ck^{-1+4\delta}=o(1).\] In particular, \(Q_z/Q_x\) is bounded above and below independently of the packet. The same comparison holds for \(Q\) throughout the ball. Since \(|\mathrm dh|\le CQ\) in the coordinate chart, \[ k|h(x+R_xY)-h(x)|\le C,\qquad |Y|\le2. \tag{41}\]

The nearest center has \(|X_z(x)|=O(k^{-10+100\delta})\). The center formula and its nearby-jet estimate in the proof of Proposition 9 show that one of its packets has \(|w(x)|\ge c e^{kh(x)}\). This proves the first line of (38).

Write \(\kappa_z=kr_0Q_z\). For every fixed \(q\ge1\), the chain rule and normal-chart bounds give \[|D_Y^qX_z|\le C_qr_0^{-1}R_x^q,\qquad \kappa_z|D_Y^qX_z| \le C_q\,\frac{Q_z}{Q_x}(kQ_x)^{1-q}\le C_q.\] For \(q=1\) there is also a lower bound: the matrix \[ \kappa_z D_YX_z=(Q_z/Q_x)D_x\xi_z \tag{42}\] has inverse bounded independently of \(k,x,z\). In particular, \[|X_z(x+R_xY)-X_z(x)| \le Ck^{-1+100\delta}/Q_x=o(k^{-1/3}).\]

Suppose first that \(w\in\mathcal T_x\). At every point where its cutoff or any derivative of the cutoff can contribute, \[\tfrac14k^{-1/3}\le |X_z|\le2k^{-1/3}.\] Apply (36). Each fixed positive \(Y\)-derivative of \(\kappa_z\Phi(X_z)\) is bounded: in a term containing \(m\) derivatives of \(X_z\), the displayed derivative bounds leave a factor \(\kappa_z^{1-m}\le1\). Amplitude derivatives are bounded as well. Derivatives of \(\omega(k^{1/3}X_z)\) are bounded because \(k^{1/3}/\kappa_z\le k^{-2/3+100\delta}\le1\). Thus differentiation introduces only fixed constants at each fixed order. Together with (41), this proves the tail bound in (38). This argument uses an absolute estimate and is valid even if \(w(x)=0\).

For \(w\in\mathcal N_x\), the displacement estimate shows that \(|X_z|<k^{-1/3}\) throughout the ball, so its cutoff is identically one. By (34), the amplitude is bounded away from zero. The bounds just proved for derivatives of \(\kappa_z\Phi(X_z)\) and \(a(X_z)\) imply bounded derivatives of \(w_x\) relative to \(|w_x|\). Integrating the bounded real logarithmic derivative along straight segments in the \(Y\)-ball gives \[|w_x(Y)|\le C|w_x(0)|=C|w(x)|.\] This proves the second line of (38), including \(j=0\).

For the lower oscillation estimate, the bounded coefficients of \(\Phi\) give \[\operatorname{Im}\partial_X\Phi(X_z(x)) =(Q'/Q_z)v+O(k^{-1/3}).\] By (30) its leading term has norm at least one. The inverse bound in (42) therefore gives a positive lower bound for the imaginary \(Y\)-gradient of \(\kappa_z\Phi(X_z)\). The amplitude contributes only \[\left|\frac{\mathrm d_Ya(X_z)}{a(X_z)}\right| \le C r_0^{-1}R_x=Ck^{-1+100\delta}/Q_x=o(1).\] It cannot cancel that lower bound. This proves (39), including at centers with \(p=0\) and for phases nearly isotropic in the complex bilinear form.

Packets whose supports do not meet the ball contribute zero to all the displayed sums. For the other packets, squaring (38), dividing by \(S_x^2\ge c e^{2kh(x)}\), and using the cardinality bound shows that the total tail contribution is at most \[C_j k^{K_1}e^{-2k^{1/4}}=o(1).\] The non-tail contributions are bounded by a constant times \(\sum_{\mathcal N_x}|w(x)|^2/S_x^2\le1\). At derivative order zero the definition of \(S_x\) also shows that this latter ratio equals \(1-o(1)\). This proves (40). ◻

Proof. We separate two deterministic regions. If \(W/e^{kh}>k^{50}\), then \(\beta=1\) for large \(k\): on its transition region (43) instead gives \(W/e^{kh}\le1+e^{-cw k}\). In the uncut region, the cosine formula and the angular derivative give \[|u_*|+|\mathrm du_*|_{g_0}\ge c e^{kb_N}.\] On \(\mathcal B_k\), the packet value and first derivative together are bounded by \(Ck^{43}e^{kh}\). The beam therefore dominates their sum on this region and gives a lower bound \(cW\) for the first jet. The first inequality in (43) puts all of \(t<s_1\) in this region for sufficiently large \(k\).

At a point in the remaining region there is a nearby net center. The persisting real jet span in (28), after converting the normalized derivative to ordinary coordinate derivatives, gives for the value and four derivatives of the random sum a covariance matrix whose least eigenvalue has square root at least \[c k^{-1}e^{kh}\ge c k^{-51}W\ge k^{-52}W\] for all sufficiently large \(k\). Only one nearby packet family is needed for this lower bound. All other families add nonnegative covariance matrices. The deterministic beam is a mean shift and does not affect the maximum of a Gaussian density. Hence, in any fixed coordinate chart and for any fixed constant \(C_0\), the unconditional probability that the normalized jet \[\mathcal J(x)=W(x)^{-1}(u(x),\partial_1u(x),\ldots,\partial_4u(x))\] has norm at most \(C_0k^{-D}\) is bounded by \[ C k^{260-5D}. \tag{51}\] Indeed the density is at most \(Ck^{260}\) and the five-dimensional ball has volume \(Ck^{-5D}\). No conditioning on \(\mathcal B_k\) is used in this estimate.

On \(\mathcal B_k\), (48) through order two and the last bound in (45) give \[|D\mathcal J|\le Ck^{45}\le k^{46}\] for sufficiently large \(k\). In each of finitely many coordinate patches covering \(B\), take a grid of mesh \(k^{-D-48}\). Its total cardinality is at most \(Ck^{4D+192}\). Apply (51) at grid points in the Gaussian region; on the other grid points the deterministic beam bound already holds on \(\mathcal B_k\). The union of the Gaussian bad events has probability at most \[C k^{260-5D+4D+192}=Ck^{452-D}.\] With \(D=500\) this tends to zero. Choose the fixed grid-point threshold \(C_0k^{-D}\) large enough to account for the finitely many coordinate norm comparisons. The interpolation error on \(\mathcal B_k\) is at most \(Ck^{-D-2}\), so the grid bounds imply \(|u|+|\mathrm du|_{g_0}\ge k^{-D}W\) everywhere. Subtracting the bad probabilities and (47) proves (50). All order-dependent constants have only increased the lower threshold on \(k\). ◻

Proof. For \(x\in V\), put \[ R_x=(kQ_x)^{-1},\qquad Q_x=(1+|\nabla_G h(x)|_G^2)^{1/2}. \tag{52}\] All balls below are Euclidean coordinate balls. Their radii are at most \(k^{-1}\), so fixed enlargements lie in a common relatively compact subset of \(\Omega\) for large \(k\). The envelope bounds give \(|DQ|\le Ck^{4\delta}\). If two candidate balls intersect, their centers are at distance at most \(2/k\), and their \(Q\) values differ by at most \(Ck^{-1+4\delta}\). Since \(Q\ge1\), the corresponding radii are uniformly comparable.

At each fixed \(k\) the candidate radii have a positive lower bound. Choose a finite maximal disjoint family of open balls \(\{B(x,R_x):x\in\mathcal X_k\}\), with centers in \(V\). Uniformly bounded enlargements of these balls cover \(V\), and \(Q\) is comparable to \(Q_x\) in each enlargement. Thus \[ \int_V kQ(y)\,\mathrm dy \le C\sum_{x\in\mathcal X_k}kQ_xR_x^4 =C\sum_{x\in\mathcal X_k}R_x^3, \qquad \sum_{x\in\mathcal X_k}R_x^3\ge c k^{1+\gamma}. \tag{53}\] The last inequality uses (19).

Fix one ball. The beam vanishes on its radius-two enlargement for large \(k\), because \(\overline V\subset\{t>1/2\}\) and \(s_4<1/2\). Define \[S_x^2=\sum_{v\in\mathcal W_k}|v(x)|^2, \qquad U_x(Y)=u(x+R_xY)/S_x,\qquad |Y|\le2.\] In expressions differentiated with respect to \(Y\), a packet \(v\) means its composition \(v(x+R_xY)\). Lemma 10 applies to the entire packet collection, including packets centered outside \(V\) that meet this ball. It gives \(S_x\ge ce^{kh(x)}\); uniformly bounded sums of normalized squared packet derivatives through every fixed order; and, for every non-tail packet, \[ \left|\operatorname{Im}\frac{\mathrm d_Yv}{v}(0)\right|\ge c. \tag{54}\] Here a non-tail has its cutoff identically one on the scaled ball and nonzero value at the center. For tails, the lemma supplies absolute exponential bounds, even if \(v(x)=0\). Polynomially many tails consequently contribute \(o(1)\) to the normalized squared value and derivative sums. In particular, \[ S_x^{-2}\sum_{v\ \mathrm{non\text{-}tail}}|v(x)|^2=1-o(1). \tag{55}\] All these statements are uniform over \(x\in\mathcal X_k\).

The value \(U_x(0)\) is a standard real Gaussian. We verify that some directional derivative has a uniformly positive conditional variance given this value. Let \(z\) be the coefficient-space vector whose coordinates for packet \(v\) are \((\operatorname{Re}v(x),\operatorname{Im}v(x))\), and let \(T\) be the matrix whose corresponding columns are the four-dimensional real derivative vectors \(\mathrm d_Y\operatorname{Re}v(0)\) and \(\mathrm d_Y\operatorname{Im}v(0)\). Thus \(|z|^2=S_x^2\). If \(g\) is the vector of independent standard Gaussian coefficients, the value and gradient are \(z^{\mathsf t}g/S_x\) and \(Tg/S_x\). The conditional gradient covariance is \[ S_x^{-2}T\left(I-\frac{zz^{\mathsf t}}{S_x^2}\right)T^{\mathsf t}. \tag{56}\] For each non-tail packet define the unit coefficient vector \(c_v\) supported on its two coordinates, with entries \[\frac{(-\operatorname{Im}v(x),\operatorname{Re}v(x))}{|v(x)|}.\] These vectors are mutually orthonormal and perpendicular to \(z\). The orthogonal projection in (56) therefore dominates \(\sum_v c_vc_v^{\mathsf t}\) as a nonnegative quadratic form. Moreover \[Tc_v=|v(x)|\operatorname{Im}\frac{\mathrm d_Yv}{v}(0).\] Equations (54) and (55) show that the trace of (56) is at least a fixed \(c>0\). There is consequently a deterministic unit direction \(e_x\) in \(\mathbb R^4\) such that the derivative \(Z_x=\partial_{e_x}U_x(0)\), conditioned on \(U_x(0)\), has variance between fixed positive constants. Its unconditional variance is bounded above by the deterministic squared-derivative estimates. Those estimates also bound the regression coefficient \(\operatorname{Cov}(Z_x,U_x(0))\).

For clarity, the needed uniform control of the local \(C^2\) norm follows from finitely many derivatives. The deterministic estimates give \[\sup_{|Y|\le2}\mathbb E|D_Y^qU_x(Y)|^2\le C_q \qquad(0\le q\le5).\] Integration and the Sobolev inequality on concentric balls in \(\mathbb R^4\) yield \[ \mathbb E\lVert U_x\rVert_{C^2(\overline{B(0,1)})}^2 \le C\mathbb E\lVert U_x\rVert_{H^5(B(0,2))}^2\le C. \tag{57}\] We choose the \(C^2\) norm to control the Euclidean gradient norm and Hessian operator norm, changing only the fixed constant in this inequality. The Sobolev embedding used here is valid since \(5>2+4/2\); see (Evans 2010, secs. 5.6–5.7).

Let \(a'=1/4\) and fix \(0<c'<a'/4\). For every sufficiently large fixed number \(H\), the Gaussian conditional density formula gives \[\mathbb P\bigl(1\le Z_x\le2,\ |U_x(0)|<c'/H\bigr)\ge c_1/H.\] Indeed, on the indicated value interval the conditional mean is bounded, and the conditional variance is bounded above and away from zero. The Gaussian density on the fixed derivative interval \([1,2]\) then has a uniform positive lower bound. By (57), failure of the bound \(\lVert U_x\rVert_{C^2}\le H\) has probability at most \(C/H^2\). Choose \(H\) once so large that the intersection of these three conditions has probability at least \(\pi_*=c_1/(2H)>0\).

On that intersection Taylor’s formula at \(\pm a'e_x/H\) gives opposite signs, with absolute values at least \(a'/(2H)\): the linear term has magnitude at least \(a'/H\), the center value has magnitude at most \(c'/H\), and the quadratic error is at most \(a'^2/(2H)\). The gradient bound preserves these signs on the parallel transverse disks of radius \[\sigma_* = a'/(4H^2)\] centered at those two points. The whole closed cylinder between the disks is compactly contained in the unit ball. The direction \(e_x\), centers and radii are deterministic functions of the packet data. After scaling by \(R_x\), they provide a potential certificate pair inside \(B(x,R_x)\).

Let \(F_k\) be the fraction of successful balls weighted by \(R_x^3\). No independence among these successes is claimed or needed. Since \(0\le F_k\le1\) and \(\mathbb EF_k\ge\pi_*\), \[ \mathbb P(F_k\ge\pi_*/2)\ge\frac{\pi_*}{2-\pi_*}>0. \tag{58}\] The Euclidean three-area of one successful disk is \(\omega_3\sigma_*^3R_x^3\), where \(\omega_3\) is the unit-ball volume in \(\mathbb R^3\). On the event in (58), (53) therefore gives a certificate of size at least \(c_*k^{1+\gamma}\). Its cylinders lie in disjoint open balls, as required by Definition 2. This proves the lemma with \(p_* =\pi_* /(2-\pi_*)\). ◻

Proof. Write \(I=g_0^{-1}\) and define globally \[ \begin{split} \mathcal D&=u^2+I(\mathrm du,\mathrm du),\qquad F=-\mathfrak r/\mathcal D,\\ \delta A&=FuI,\qquad \delta\rho=k^{-2}\bigl(Fu-\mathop{\mathrm{div}}_\mu(FI\,\mathrm du)\bigr). \end{split} \tag{60}\] The lower bound on the first jet gives \(\mathcal D\ge\tfrac12 k^{-1000}W^2>0\), so these are smooth. The product rule gives the exact identity \[\begin{align*} \mathop{\mathrm{div}}_\mu(\delta A\,\mathrm du)+k^2\delta\rho\,u &=u\mathop{\mathrm{div}}_\mu(FI\,\mathrm du)+F I(\mathrm du,\mathrm du)\\ &\quad+Fu^2-u\mathop{\mathrm{div}}_\mu(FI\,\mathrm du)\\ &=F\mathcal D=-\mathfrak r. \end{align*}\] Thus \(\mathop{\mathrm{div}}_\mu((A+\delta A)\mathrm du) +k^2(\rho+\delta\rho)u=0\) identically.

We include finite derivative bounds to specify the accuracy needed in this step. For \(0\le q\le R+1\), the hypotheses imply \[|D^q\mathcal D|\le C_q k^{2q+86}W^2.\] Every term in a derivative of order \(q\) of \(\mathcal D^{-1}\) is a constant times \(\mathcal D^{-m-1}\prod_{i=1}^m D^{q_i}\mathcal D\), where \(q_i\ge1\), \(\sum_iq_i=q\), and \(m\le q\); the order-zero case is included separately. Hence \[|D^q(\mathcal D^{-1})| \le C_q k^{1000+2q+1086q}W^{-2} =C_q k^{1000+1088q}W^{-2}.\] By the product rule, \[ |D^qF|\le C_qk^{-M+1000+1088q}W^{-1} \qquad(0\le q\le R+1). \tag{61}\] No derivatives of \(W^{-1}\) are required for these estimates: the displayed weights bound each differentiated factor before the factors are multiplied.

Multiplying (61) by derivatives of \(u\) gives \(|D^q(Fu)|\le C_qk^{-M+1041+1088q}\). Replacing \(u\) by \(I\mathrm du\) gives the bound \(C_qk^{-M+1043+1088q}\), with the derivatives of the fixed tensor \(I\) absorbed in the constant. The divergence in (60) costs one further derivative, and the density correction has the factor \(k^{-2}\). These bounds give (59); its exponent for the density is \(1043+1088(R+1)-2=2129+1088R\). The cancellation of the two weights is what permits arbitrarily small coefficient changes even where the original function is exponentially small.

If \(\mathfrak r=0\) on an open set, then \(F=0\) there and both corrections vanish. Finally, the cone of positive pairs is open in \(C^0\) on the compact base, so sufficiently small corrections preserve positivity. ◻

Proof. Take a small product coordinate patch \(O\) in the stated region on which \((u,t)\) are the first two coordinates, and \(\mathrm du\ne0\). Choose a smooth nonnegative bump \(\chi\) supported in \(O\) and strictly positive on a smaller product patch \(O_0\). Define \[E=\chi\left(I- \frac{(I\mathrm du)\otimes(I\mathrm du)}{I(\mathrm du,\mathrm du)}\right), \qquad I=g_0^{-1},\] extending by zero off \(O\). This is a smooth nonnegative symmetric contravariant tensor and \(E\mathrm du=0\).

Let \(\mathcal E\) be the real eigenspace at \(k^2\) for the given pair. The nonnegative form \[f\longmapsto\int_B E(\mathrm df,\mathrm df)\,\mathrm d\mu, \qquad f\in\mathcal E,\] has kernel exactly \(\operatorname{span}\{u\}\). Indeed, a kernel element satisfies \(\mathrm df\in\operatorname{span}\{\mathrm du\}\) on \(O_0\). After making the product patch smaller, this means \(f=b(u)\) for a smooth one-variable function \(b\). The round eigenfunction equations imply \[ b''(u)|\mathrm du|_{g_0}^2=k^2\bigl(ub'(u)-b(u)\bigr). \tag{62}\] The Euclidean gradient of the homogeneous harmonic polynomial defining \(u_*\) has squared norm \(N^2(1-t)^{N-1}\), and its radial derivative is \(Nu\). Tangential projection onto \(S^4\) consequently gives \[|\mathrm du|_{g_0}^2=N^2\bigl((1-t)^{N-1}-u^2\bigr).\] Differentiate (62) with respect to \(t\) while fixing the coordinate \(u\). Since \(N>1\) and \(t<1\), it follows that \(b''(u)=0\). Substitution back into (62) eliminates the constant term of \(b\), so \(f=cu\) on \(O_0\).

The difference \(f-cu\) satisfies a second-order elliptic equation with smooth real positive definite principal coefficients and smooth lower-order coefficients. Aronszajn’s unique continuation theorem applies: its local hypotheses allow principal coefficients of class \(C^2\) with Lipschitz second derivatives and bounded lower-order coefficients, and smooth solutions satisfy its solution hypotheses (Aronszajn 1957, 235–36). Vanishing on an open patch gives infinite-order vanishing there. On the connected base the difference is therefore identically zero, proving the assertion about the kernel.

For \(\tau>0\) replace \(A\) by \(A+\tau E\). The eigenfunction \(u\) and its eigenvalue remain exact because \(E\mathrm du=0\) identically. We claim that \(k^2\) is simple for every sufficiently small positive \(\tau\). Otherwise there would be \(\tau_m\downarrow0\) and real eigenfunctions \(f_m\) at \(k^2\), normalized in \(L^2(\rho\,\mathrm d\mu)\) and orthogonal to \(u\). Their energy identity is \[\int_B(A+\tau_m E)(\mathrm df_m,\mathrm df_m)\,\mathrm d\mu=k^2,\] so uniform ellipticity bounds them in \(H^1\). A subsequence converges weakly in \(H^1\) and strongly in \(L^2\) to a normalized function \(f_0\) orthogonal to \(u\). Passing to the weak equation shows \(f_0\in\mathcal E\); elliptic regularity makes it smooth. For any fixed \(\phi\in\mathcal E\), testing the two eigenfunction equations against one another cancels the original form and gives \[\int_B E(\mathrm df_m,\mathrm d\phi)\,\mathrm d\mu=0.\] Weak convergence passes this fixed bilinear identity to the limit. Now choose \(\phi=f_0\) to see that \(f_0\) is in the kernel just identified, contradicting its normalization and orthogonality. This proves the claim. The tensor \(E\) is fixed before choosing \(\tau\); hence \(\tau E\) can meet every prescribed finite norm tolerance, while remaining supported in \(O\). ◻

Proof of Proposition 3. Fix the input pair, the collar, \(r\) and \(\eta\). Choose \(R\ge\max\{r,1\}\) and a smaller positive tolerance \(\eta'\) so that total \(C^R\) movement less than \(\eta'\) implies the requested \(C^r\) movement and keeps the pair positive and in \(\mathcal U\). Such a choice exists because the fixed input belongs to the open set \(\mathcal U\).

Choose \(J=R+2\) and \(M\ge2131+1088R\), and then choose the finite packet orders giving (48). All these choices precede frequency selection. Lemmas 11 and 12 give, for every sufficiently large allowed \(k\), a sample with the required first-jet bound, residual bounds and a certificate of size \(c_*k^{1+\gamma}\). Lemma 13 changes the pair by at most \(C_Rk^{-2}\) in \(C^R\) while preserving that same function. Choose \(k\) large enough for this change to be less than \(\eta'/2\).

We describe explicitly why a narrow collar only changes this frequency threshold. Packet supports avoid the region below \(s_1-Ck^{-1/3-100\delta}\). Requiring \[ Ck^{-1/3-100\delta}<w/100, \qquad cw k>(M'+P)\log k+\log C_q(w) \tag{63}\] protects a fixed subcollar and makes any required cutoff residual \(C_q(w)k^Pe^{-cw k}\) smaller than \(k^{-M'}\). Only finitely many orders \(q\) and finite targets \(M'\) are needed at this step. For every already fixed \(w>0\), all inequalities in (63) hold for every sufficiently large \(k\). In particular the exact corrections vanish below \(s_1-w/100\), where the coefficients remain round and \(u=u_*\).

In the subcollar \[l_-+w/20<t<l_-+w/10\] choose a point with \(\sin(N\vartheta)\ne0\). Since \(0<t<1\), \(\mathrm dt\ne0\), and the nonzero angular derivative makes \(\mathrm du\) and \(\mathrm dt\) independent. Lemma 14 applies there. After \(k\) and the sample are fixed, form its tensor \(E\) and choose \(\tau>0\) below its simplicity threshold and so small that \(\lVert\tau E\rVert_{C^R}<\eta'/2\). This tensor can have large finite derivatives depending on \(k\); the subsequent scalar choice of \(\tau\) absorbs them. It preserves \(u\), its exact eigenvalue and all current signs exactly. Its support lies in \(t>l_-\), so the final pair is round on \(t<l_-\).

The final pair meets every conclusion of Proposition 3. In particular, its certificate constant \(c_*\) was fixed after choosing the finite orders and before taking \(k\) large, and it was unaffected by either correction. The exponent \(\gamma\), the chart and the background neighborhood were fixed upstream. Neither the collar estimates nor the arbitrary finite tolerances change them. The construction works for every sufficiently large integer \(N\), as asserted. ◻

Proof. We recall the spectral and regularity facts used here. The form \[\int_B A(\mathrm df,\mathrm dv)\,\mathrm d\mu+ \int_B\rho fv\,\mathrm d\mu\] is coercive on \(H^1(B)\) and defines the inverse of \(-L+1\) in \(L^2(\rho\,\mathrm d\mu)\). The compact embedding \(H^1\hookrightarrow L^2\) makes this inverse compact, and it is selfadjoint. The spectral theorem and elliptic regularity give a discrete nonnegative spectrum with finite multiplicities and smooth real eigenfunctions. Its ordered eigenvalues obey the min–max principle with Rayleigh quotient \[ \frac{\int_B A(\mathrm df,\mathrm df)\,\mathrm d\mu} {\int_B\rho f^2\,\mathrm d\mu}. \tag{64}\]

Uniform multiplicative comparisons of nearby positive tensors and densities compare (64) for every \(f\) and therefore imply convergence of each ordered eigenvalue under \(C^0\) convergence of pairs. Choose a smaller interval about \(\lambda\), with closure in \(I\), separating it from its neighboring ordered eigenvalues. Their convergence implies that for sufficiently close pairs the corresponding ordered eigenvalue is still simple and lies in \(I\).

We also need uniform convergence of a corresponding normalized eigenfunction, up to sign. Here only finitely many coefficient derivatives are needed. For a fixed smooth elliptic \(L\) on the closed base, the local interior estimates, a finite partition of unity and interpolation give \[ \lVert f\rVert_{H^{a+2}} \le C_a\bigl(\lVert Lf\rVert_{H^a}+\lVert f\rVert_{L^2}\bigr) \qquad(a=0,1,2,\ldots). \tag{65}\] These are the usual interior elliptic estimates; the Sobolev and compactness statements used here are described in (Evans 2010, secs. 5.6–5.7 and 6.3). A sufficiently small finite coefficient norm makes \(L'-L\) small as an operator \(H^{a+2}\to H^a\) for each of finitely many specified \(a\). Its contribution in (65) can then be absorbed on the left. Applying the resulting estimates at \(a=0,2\), together with the eigenfunction equation, gives a uniform \(H^4\) bound for eigenfunctions whose eigenvalues stay in the chosen bounded interval and whose \(L^2\) norms are bounded. For example, requiring sufficiently small \(C^4\) changes of the pair is more than enough for these finitely many operator bounds: the principal coefficients of \(L\) involve \(A,\rho\) without derivatives, and its first-order coefficients involve at most their first derivatives.

Suppose that pairs converge in such a finite norm to \(\mathcal P\). Normalize their corresponding real eigenfunctions \(f_m\) by \(\int\rho_m f_m^2\,\mathrm d\mu=1\). Positivity bounds their ordinary \(L^2\) norms, so the preceding argument bounds them in \(H^4\). Rellich compactness and Sobolev embedding in dimension four give a subsequence converging in \(C^0\) (for example, first converge in \(H^3\), which embeds in \(C^0\)). Passing to the weak equation shows that the limit is a normalized eigenfunction at \(\lambda\) for \(\mathcal P\). By simplicity it equals the prescribed normalized eigenfunction or its negative.

The finite collection of strict signs on compact disks has a positive minimum absolute margin after this normalization. Hence all sufficiently close such eigenfunctions have the same opposite signs on the same disks, allowing an overall change of sign. If no sufficiently small finite-norm neighborhood had this property, a sequence converging in that norm would contradict the preceding compactness argument. Finally, decrease its radius to put a closed finite-norm ball inside the neighborhood. This proves the lemma, including the assertion for varying densities. ◻

Proof of Theorem 1. Use monotone fixed coordinate norms for pairs. Choose \(b>0\) so that the closed \(C^1\) ball of radius \(b\) about \(\mathcal P_0=(g_0^{-1},1)\) consists of positive pairs in \(\mathcal U\). Fix \[l^{(j)}=\frac14+2^{-j-2}\quad(j\ge0),\qquad \epsilon_0=\gamma/4>0.\] We construct pairs \(\mathcal P_j=(A_j,\rho_j)\), frequencies \(k_j\), and certificates \(\mathcal C_j\) inductively. Every output is round on \(t<l^{(j)}\).

After producing \(\mathcal P_i\), apply Lemma 15 to its simple eigenvalue \(k_i^2\), its certificate, and the interval \((k_i^2/2,2k_i^2)\). Denote a resulting finite norm order and closed-ball radius by \(s_i\) and \(\varepsilon_i>0\). At stage \(j\), these numbers are already fixed for \(i<j\). Put \[ \begin{split} r_j&=\max\{j+1,1,s_1,\ldots,s_{j-1}\},\\ \eta_j&=\min\left\{2^{-j},\ b2^{-j-1},\, \min_{1\le i<j}2^{-(j-i+1)}\varepsilon_i\right\}. \end{split} \tag{66}\] For \(j=1\), the inner minimum is omitted. These are finite requirements at each stage. Apply Proposition 3 to \(\mathcal P_{j-1}\), with \(l_+=l^{(j-1)}\), \(l_-=l^{(j)}\), order \(r_j\) and tolerance \(\eta_j\). Use the certificates from its construction in Lemma 12, with balls centered in \(V\) and radii at most \(k_j^{-1}\). The proposition gives a positive constant \(c_j\) and all sufficiently large allowed frequencies. Choose one with \(k_j>2k_{j-1}\) when \(j>1\), and also so large that \[c_jk_j^{\gamma/2}>j.\] The output certificate then satisfies \[ |\mathcal C_j|>j k_j^{1+\gamma/2}. \tag{67}\] Only after this output has been constructed do we select its own \(s_j,\varepsilon_j\) by Lemma 15. They constrain subsequent stages, so no current frequency choice depends circularly on its own future persistence radius.

The increments are summable in every fixed \(C^r\) norm: for all large \(j\), \(r_j\ge r\) and their \(C^{r_j}\) norms are less than \(2^{-j}\). Therefore the pairs converge to one smooth pair \(\mathcal P_\infty=(A_\infty,\rho_\infty)\). Their total \(C^1\) movement from the round pair is less than \(b\sum_{j\ge1}2^{-j-1}=b/2\), so the limit remains positive and belongs to \(\mathcal U\). For each fixed \(i\), the same budgets give \[\lVert\mathcal P_\infty-\mathcal P_i\rVert_{C^{s_i}} \le\sum_{j>i}2^{-(j-i+1)}\varepsilon_i =\varepsilon_i/2.\] The limit lies in every selected persistence neighborhood. Hence there are real smooth eigenfunctions \(v_i\) for the limiting base operator, with the same certificate disks \(\mathcal C_i\), and \[ -L_\infty v_i=\lambda_i v_i,\qquad k_i^2/2<\lambda_i<2k_i^2. \tag{68}\] The frequency separation makes these eigenvalue intervals disjoint and tending to infinity. The functions \(v_i\) are nonzero by their normalization, or just by their strict signs.

Set \(G_\infty^{-1}=A_\infty/\rho_\infty\) and let \[q=\frac{\rho_\infty\,\mathrm d\mu}{\mathrm d\operatorname{vol}_{G_\infty}}.\] This is a smooth positive function, the ratio of two smooth positive base densities. We use the standard warped-product realization of a weighted Laplacian (Charalambous and Lu 2015, sec. 4, equation (2)), with a circle as the fiber. On the fixed manifold \(M=B\times S^1\) define \[ g=G_\infty+q^2\mathrm d\varphi^2, \tag{69}\] where \(\mathrm d\varphi^2\) is the standard unit-circle metric. The manifold is smooth, closed and connected, of dimension five, and \[\mathrm d\operatorname{vol}_g =q\,\mathrm d\operatorname{vol}_{G_\infty}\,\mathrm d\varphi =\rho_\infty\,\mathrm d\mu\,\mathrm d\varphi.\] For a function \(f\) pulled back from \(B\), the coordinate divergence formula gives \[ \Delta_g f =\frac{1}{\rho_\infty\mu} \partial_a\bigl(\rho_\infty\mu G_\infty^{ab}\partial_b f\bigr) =\rho_\infty^{-1}\mathop{\mathrm{div}}_\mu(A_\infty\mathrm df) =L_\infty f. \tag{70}\] Here \(\mu\) in the coordinate expression denotes its positive coordinate density. Thus the pullbacks \(u_i\) of \(v_i\) are exact real Laplace–Beltrami eigenfunctions with eigenvalues \(\lambda_i\).

It remains to convert their certificates into the claimed measure bound. Fix a closed interval \(I_0\) of positive length in a circle coordinate chart. All certificate cylinders are in one fixed compact subset of \(\Omega\): their centers lie in \(V\Subset\Omega\) and their radii tend uniformly to zero, with the initial frequencies chosen large enough for the same inclusion. On the product of this compact set with \(I_0\), coordinate and Riemannian distances have fixed Lipschitz comparison constants. One may see this directly by extending the coordinate functions smoothly with cutoffs to the compact manifold; their gradients are then uniformly bounded in the single fixed metric \(g\).

For one certificate cylinder \(C_i^{(a)}\), let \(D_i^{(a)}\) be one of its transverse disks. The opposite endpoint signs imply that every longitudinal segment through the disk contains a zero of \(v_i\). Therefore the subset \[\mathcal Z(u_i)\cap(C_i^{(a)}\times I_0)\] projects onto \(D_i^{(a)}\times I_0\) by coordinate orthogonal projection in the base and the circle coordinate. The Lipschitz constant of every such projection is bounded by one fixed \(K\), independent of the disk’s radius and orientation. The defining covering inequality for Hausdorff measure under a Lipschitz map gives \[\mathcal H_g^4\bigl(\mathcal Z(u_i)\cap(C_i^{(a)}\times I_0)\bigr) \ge K^{-4}\mathcal H_{\mathrm{Euc}}^4(D_i^{(a)}\times I_0) =K^{-4}|I_0|\mathcal H_{\mathrm{Euc}}^3(D_i^{(a)}).\] The closed cylinders lie in disjoint open balls. Their indicated nodal subsets are consequently pairwise disjoint Borel sets, so their Hausdorff measures add. Using (67), we obtain one constant \(c>0\) belonging to the final metric such that \[ \mathcal H_g^4(\mathcal Z(u_i))\ge c|\mathcal C_i| > c i k_i^{1+\gamma/2}. \tag{71}\] This argument uses continuity and signs only, without any regularity assumption on the nodal set.

Finally, (68) and \(\epsilon_0=\gamma/4\) imply \[\lambda_i^{1/2+\epsilon_0} \le2^{1/2+\epsilon_0}k_i^{1+\gamma/2}.\] Combining this with (71) yields \[\frac{\mathcal H_g^4(\mathcal Z(u_i))}{\lambda_i^{1/2+\epsilon_0}} \ge c\,2^{-1/2-\epsilon_0}i\longrightarrow\infty.\] Both \(g\) and \(\epsilon_0\) are fixed throughout this sequence, which proves Theorem 1. ◻

Proof. Take the product of (69) with any fixed smooth closed connected manifold of dimension \(n-5\), and pull back the same eigenfunctions. The product Laplacian preserves their eigenvalues. Multiply each of the projection certificates above by a fixed compact coordinate box of positive \((n-5)\)-volume in the extra factor. The same Lipschitz projection argument, now in dimension \(n-1\), gives a fixed positive multiple of the preceding certificate lower bound. The denominator and its fixed exponent are unchanged, so the ratio still tends to infinity. ◻