Proof. For the first assertion, choose a compact subset of \(U\) carrying more than three quarters of its mass. A finite subcover of the prescribed neighborhoods has a Lebesgue number on this compact set. A sufficiently fine coordinate grid therefore gives finitely many cubes, each with closure in one of the prescribed neighborhoods and in \(U\), whose union contains that compact set except for grid faces. The faces have zero mass. Shrinking the cores inside these cubes loses arbitrarily little mass, and smooth cutoffs supported in the cubes can be chosen to equal one on the cores.

For the second assertion, submersion coordinates and a finite partition of unity express the pushforward of \(\mu|_C\) under \(y\) as an integrable density on \(\mathbb R^q\). Thus it suffices to average the bounded periodic function \(\mathbf 1_E(N\cdot)\) against an \(L^1\) function. For the indicator of a rectangle, the assertion follows by tiling with period cells and estimating the cells meeting the rectangle’s boundary. Approximation of an \(L^1\) function by finite linear combinations of rectangle indicators proves the claim. The boundary assumption on \(E\) ensures that its cell average is well defined and unchanged by choices on its boundary. ◻

Proof. Put \(a=a_\phi\). At each point, choose a strict unit direction \(v\perp a\). Because \(a\) does not vanish, the flow-box construction gives a local first integral \(s\), with \(\,\mathrm ds(a)=0\), whose differential at the point is any prescribed nonzero covector annihilating \(a\). Prescribe it also to annihilate \(v\). On a sufficiently small neighborhood, the line \[a^\perp\cap\ker\,\mathrm ds\] continues to be strict, with a positive margin on a smaller compact neighborhood. The differential \(\,\mathrm ds\) remains nonzero there.

Apply Lemma 5 to these neighborhoods with \(\,\mathrm d\mu=|a|\,\,\mathrm dV_g\). It gives finitely many cubes and cutoff-one cores capturing at least half the mass. We first describe the perturbation on one cube, suppressing its index. Let \(\chi\) be its cutoff. For a constant \(C>0\) to be fixed before \(N\), set \[\psi_N=\phi+C N^{-2}\chi h(Ns).\] Direct differentiation gives, uniformly on the patch, \[\begin{align*} a_{\psi_N} &=a+\frac C N\chi h'(Ns)\mathop{\mathrm{grad}}s +\frac C{N^2}h(Ns)\mathop{\mathrm{grad}}\chi,\\ H_{\psi_N} &=H_\phi+C\chi h''(Ns)\,\,\mathrm ds\otimes\,\mathrm ds+O(N^{-1}). \end{align*}\] All constants in this proof may depend on the fixed patch data, including \(C\), but not on \(N\).

For large \(N\), choose a unit vector \(v_N\) on the line \[a_{\psi_N}^{\perp}\cap\ker\,\mathrm ds.\] This line tends uniformly to the old strict line. The added leading Hessian has zero value on \(v_N\), while \(\,\mathrm ds(a_{\psi_N})=O(N^{-1})\). Its value on the new gradient is therefore \(O(N^{-2})\), even on the portions where \(h''\) is negative. The remaining Hessian error is \(O(N^{-1})\). The old strict margin consequently preserves strictness for all sufficiently large \(N\).

We choose \(C\) to obtain fullness on the cutoff-one core where \(Ns\in E_h\). For the old gradient, consider the symmetric form on \(a^\perp\) \[B=(1+|a|^2)H_\phi|_{a^\perp}+H_\phi(a,a)\mathop{\mathrm{Id}}.\] In an orthonormal basis \((v,w)\) with \(\,\mathrm ds(v)=0\) and \(w=\mathop{\mathrm{grad}}s/|\,\mathrm ds|\), its \(vv\) entry is bounded below by some \(m>0\). All its entries are bounded on the patch. Adding \[C(1+|a|^2)|\,\mathrm ds|^2\,w\otimes w\] makes \(B\) uniformly positive definite when \(C\) is sufficiently large. Indeed, if \(|B_{vw}|\le M\) and \(B_{ww}\ge-M\), it is enough to make the new \(ww\) entry exceed \(M^2/m+m\); the Schur complement is then positive. The factor \((1+|a|^2)|\,\mathrm ds|^2\) has a positive patch minimum, so a finite \(C\) suffices. On the specified core \(\chi=1\) and \(h''>1\), so the actual leading addition is at least this large. Changing the gradient by \(O(N^{-1})\) and adding the Hessian remainder preserves its positive minimum for large \(N\). Thus \(\psi_N\) is full there.

Perform these perturbations on the finite disjoint patches, with their chosen constants \(C\), using a common sufficiently large \(N\). The changes are arbitrarily small in \(C^1\), preserve nonvanishing, and have compact support in \(\Omega\). Lemma 5 shows that the union \(G_N\) of the cutoff-one regions with \(Ns\in E_h\) satisfies \[\int_{G_N}|a|\,\,\mathrm dV_g \ge\frac{p_h}{4}\int_\Omega|a|\,\,\mathrm dV_g\] for sufficiently large \(N\). Choose \(N\) still larger so that the stated pointwise relative gradient bound holds. With \(\mathcal M(\zeta)=\int_\Omega|a_\zeta|\,\,\mathrm dV_g\), the full region therefore has mass at least \[(1-\epsilon_p)\frac{p_h}{4}\mathcal M(\phi) \ge\frac{p_h(1-\epsilon_p)}{4(1+\epsilon_p)}\mathcal M(\psi_N) \ge\frac{p_h}{8}\mathcal M(\psi_N).\] Fullness is open where the gradient is nonzero, so this gives the asserted open full region. The \(C^0\) bound and any additional prescribed \(C^1\) smallness follow by increasing \(N\) once more. ◻

Proof. Put \(a=a_\psi\) and \(a_N=a_{\Phi_N}\). When \(Ny\) lies in a periodic disk, let \(d_R,d_T\) be the radial and angular unit-coefficient combinations of \(\,\mathrm dy_1,\,\mathrm dy_2\). The singular-value hypothesis gives \[\frac12\le |d_R|,|d_T|\le2.\] These covectors need not be orthogonal in the metric. Outside the disks, set the leading derivatives \(f',f'',f'/s\) to zero. There, or in the constant central portions of the disks, choose any orthonormal coefficient pair when such notation is needed. Set \(f'/s=0\) at the center. Since \(f\) is constant near the center, this agrees with its smooth continuation.

The exact gradient formula and the Hessian expansion are \[ \begin{aligned} a_N&=a+q\,d_R^\sharp+e_N,\\ q&=\delta A\chi f',\qquad e_N=\frac{\delta A}{N}F(Ny)\mathop{\mathrm{grad}}\chi, \end{aligned} \tag{6}\] \[ H_{\Phi_N} =H_\psi+N\delta A\chi \left[f''d_R^2+\frac{f'}s d_T^2\right]+E_N, \tag{7}\] with \[ |E_N|\le C_P\left((\chi+|\,\mathrm d\chi|)f'+N^{-1}\right). \tag{8}\] Here and below \(C_P\) denotes a finite constant depending on the fixed patch data, independent of \(N\). To check the remainder, its terms are \[\begin{align*} &\delta A\chi\sum_i F_i(Ny)\mathop{\mathrm{Hess}}y_i,\\ &\delta A\left(\,\mathrm d\chi\otimes\sum_i F_i(Ny)\,\mathrm dy_i +\sum_i F_i(Ny)\,\mathrm dy_i\otimes\,\mathrm d\chi\right),\\ &\frac{\delta A}{N}F(Ny)\mathop{\mathrm{Hess}}\chi. \end{align*}\] Since \(|(F_1,F_2)|=f'\), these give Equation (8). In the constant portions of \(F\), its value contributes only to \(e_N\) and to the last displayed term.

We first control the leading Hessian trace in (7). The leading increment in (6) is perpendicular to \(a\), so \[|a_N|\ge |a|-|e_N|.\] Thus \(a_N\) is nonzero for large \(N\). Choose a vector \(b_N\) satisfying \[b_N\perp a_N,\qquad d_R(b_N)=0,\qquad |b_N|=L_N:=(1+|a_N|^2)^{1/2}.\] The two linear constraints have a nonzero common solution in dimension three. They imply \[\langle a,b_N\rangle=-\langle e_N,b_N\rangle.\] Because \(|a|\) has a positive patch minimum, the component of \(b_N/L_N\) along \(a\) tends uniformly to zero. The kernel of \(\,\mathrm dy\) is precisely the line spanned by \(a\). Its singular-value lower bound and the identity \(d_R(b_N)=0\) consequently give, for all sufficiently large \(N\), \[|d_T(b_N)|=|\,\mathrm dy(b_N)|\ge\frac14 L_N.\] Also \(A\le3L_N\) for large \(N\), and \[|d_R(a_N)|\le4\delta A\chi f'+2|e_N|.\] With \(K=\|f''\|_\infty\) and \(B_N=f''d_R^2+(f'/s)d_T^2\), it follows that \[\begin{align*} B_N(a_N,a_N)+B_N(b_N,b_N) &\ge\frac1{16}L_N^2\frac{f'}s-K|d_R(a_N)|^2\\ &\ge\frac1{16}L_N^2\frac{f'}s -288K\delta^2L_N^2(f')^2-8K|e_N|^2. \end{align*}\] The estimate uses \((x+y)^2\le2x^2+2y^2\); in particular, its final error is \(O_P(N^{-2})\). Write \(S_0=\sup_s sf'(s)>0\). Choose once and for all \[0<\delta^2\le\frac1{9216KS_0}.\] Then \((f')^2\le S_0 f'/s\) gives \[ \begin{split} N\delta A\chi \bigl[B_N(a_N,a_N)+B_N(b_N,b_N)\bigr] \ge N\delta A\chi\frac{L_N^2}{32}\frac{f'}s -C_P N^{-1}. \end{split} \tag{9}\] All constants used to choose \(\delta\) depend only on \(F\) and the numerical singular-value bounds.

We next control the transition where the favorable leading term can be small. A nonnegative smooth cutoff satisfies \[|\,\mathrm d\chi|\le C_P\sqrt\chi.\] Indeed, extend it by zero in the coordinate chart. If its Euclidean Hessian is bounded by \(M>0\), the Taylor inequality at \(x-M^{-1}D\chi(x)\) gives \(|D\chi(x)|^2\le2M\chi(x)\); metric equivalence gives the stated bound. Consequently, with \(t=\chi f'\), Equations (6) and (8) imply \[ |a_N-a|\le C_P(t+N^{-1}),\qquad |E_N|\le C_P(t+\sqrt t+N^{-1}). \tag{10}\]

Let \(m>0\) be the minimum of the full trace \(\mathcal T(H_\psi,a;v)\) over the transverse unit circles on \(\overline P\). Project \(b_N/L_N\) onto \(a^\perp\) and normalize; for large \(N\) this gives a unit vector \(v_N\perp a\) differing from \(b_N/L_N\) by \(O_P(N^{-1})\). Using also \(|L_N-L_\psi|\le|a_N-a|\), the base Hessian trace is at least \[H_\psi(a_N,a_N)+H_\psi(b_N,b_N) \ge m-C_P(t+N^{-1}).\] The vectors \(a_N,b_N\) are uniformly bounded for this fixed patch, so the trace of the remainder in (10) is bounded in absolute value by \(C_P(t+\sqrt t+N^{-1})\). Choose \(t_0>0\) sufficiently small that these terms involving \(t\) and \(\sqrt t\) total less than \(m/4\) when \(t\le t_0\). For sufficiently large \(N\), Equation (9) now gives a strictly positive total trace throughout \(t\le t_0\).

On the complementary region \(t\ge t_0\), one has \(s\le R\), and the positive term in (9) is at least \[\frac{N\delta A t_0}{32R}.\] It tends to infinity and dominates the bounded base and remainder traces. This proves strictness for all sufficiently large \(N\). In this argument \(t_0\) and the frequency threshold can depend on the full margin \(m\) and all patch derivatives. The choice of \(\delta\) is already fixed. In particular, the region \(t\to0\) requires no lower bound for the product \(Nt\).

Finally, exact orthogonality in the leading gradient gives \[|a+q d_R^\sharp|^2=|a|^2+q^2|d_R|^2.\] The uniform error \(e_N=O_P(N^{-1})\) proves (4). Where \(\chi=1\) and \(Ny\in E_F\), the bounds \(A\ge|a|/2\) and \(|d_R|\ge1/2\) give \[|a+q d_R^\sharp| \ge |a|\sqrt{1+\frac{\delta^2c_1^2}{16}}.\] Set \[g_0=\sqrt{1+\frac{\delta^2c_1^2}{16}}-1>0.\] Decreasing the fixed \(\delta\) if necessary, assume \(g_0\le1\). The positive minimum of \(|a|\) allows \(|e_N|\le\epsilon_r|a|\) for large \(N\), proving (5). The \(C^0\) convergence follows directly from the factor \(N^{-1}\) in the definition of \(\Phi_N\). ◻

Proof of Proposition 4. Keep the periodic functions and the constants from Lemma 7 fixed. Define \[\theta=\frac{p_h p_F}{64},\qquad t_* =\theta g_0,\qquad \eta=\frac{t_*}{4}>0.\] We prove the cycle statement, allowing any prescribed positive \(C^0\) budget for that cycle.

Apply Lemma 6 with \(\epsilon_p\le\min(1/3,t_*/16)\) and half the \(C^0\) budget. Write \(\psi\) for the resulting profile and \(U\) for its full region. The preparation, including its frequency, is now held fixed. On a sufficiently small neighborhood of each point of \(U\), the flow-box construction gives two first integrals \(y_1,y_2\) of \(a_\psi\). Their differentials can be prescribed to form an orthonormal basis of \(a_\psi^\perp\) at the center; after shrinking, the nonzero singular values of \(\,\mathrm dy\) lie in \([1/2,2]\). One can also choose a constant \(A\) on that patch with \(1/2\le A/|a_\psi|\le2\).

Apply Lemma 5 in \(U\) with density \(|a_\psi|\,\,\mathrm dV_g\). Its radial patch cores capture at least half the full-region mass. In view of Equation (3), their total mass is at least \((p_h/16)\mathcal M(\psi)\). For all sufficiently large common radial frequencies \(N\), periodic averaging gives a set of cutoff-one annular points of mass at least \[\frac{p_h p_F}{32}\mathcal M(\psi) \ge\theta\mathcal M(\psi).\] Use Lemma 7 on these disjoint patches. Increase \(N\) to obtain strictness, nonvanishing, the remaining half of the \(C^0\) budget, and relative errors \[\epsilon_r\le\frac{t_*}{8},\qquad \epsilon_r\le\frac{g_0}{2}.\] Call the resulting profile \(\widehat\psi\). On the selected annular set the gradient multiplier is at least \(1+g_0/2\); everywhere else it is at least \(1-\epsilon_r\). Therefore \[\begin{align*} \mathcal M(\widehat\psi) &\ge\left(1-\epsilon_r+\frac{\theta g_0}{2}\right) \mathcal M(\psi)\\ &\ge\left(1+\frac{3t_*}{8}\right)\mathcal M(\psi)\\ &\ge\left(1-\frac{t_*}{16}\right) \left(1+\frac{3t_*}{8}\right)\mathcal M(\phi)\\ &\ge(1+\eta)\mathcal M(\phi). \end{align*}\] The last inequality uses \(0<t_*\le1\). This establishes a gain independent of the incoming profile and of the full-region patch sizes. The latter are fixed only after preparation; their finite bounds determine how large the radial frequency must be.

Since \(\mathcal M(\phi)>0\), finitely many cycles raise it above \(T\). Divide \(\varepsilon\) among those cycles. Each cycle has compact support in \(\Omega\) and preserves strictness and nonvanishing on \(\overline\Omega\). Their finite union of supports is compactly contained in \(\Omega\), and their total \(C^0\) change is less than \(\varepsilon\). All functions produced are smooth. This completes the proof of Proposition 4. ◻

Proof. The function \(\phi_0\) is smooth on the sphere. Its round Hessian is \[ \mathop{\mathrm{Hess}}_{g_*}\phi_0 =2(\,\mathrm dx_1^2+\,\mathrm dx_2^2)-2r^2g_*. \tag{12}\] The eigenvalues of \(\,\mathrm dx_1^2+\,\mathrm dx_2^2\) on the tangent three-plane are \(1\), \(1-r^2\), and \(0\): its two nonzero eigenvalues are those of the Gram matrix \(\mathop{\mathrm{Id}}_2-(x_1,x_2)^T(x_1,x_2)\). The Hessian eigenvalues in (12) are consequently \[2-2r^2,\qquad 2-4r^2,\qquad -2r^2.\] On \(r\le3/10\), the first two exceed \(1\), and the third exceeds \(-1/2\). These inequalities persist uniformly in a sufficiently small \(C^2\) neighborhood \(\mathcal U\) of the round metric. The derivatives of \(\phi_0\) do not depend on \(t\), so the same neighborhood works for all the stated \(t\).

For such a metric, intersect a positive two-plane of \(H_{\phi_0}\) with \(a_{\phi_0}^\perp\) and choose a unit vector \(v\) in the intersection. Then \[\mathcal T(H_{\phi_0},a_{\phi_0};v) \ge-\frac12|a_{\phi_0}|^2+1+|a_{\phi_0}|^2>0.\] Thus \(\phi_0\) is strict throughout \(r\le3/10\). Also \[|\mathop{\mathrm{grad}}_{g_*}(r^2)|^2=4r^2(1-r^2).\] Vanishing of a differential is independent of the metric, so \(\phi_0\) has nonvanishing gradient for \(0<r<1\), in particular on a neighborhood of \(\overline\Omega\).

For the envelope comparison, put \[j_t(r)=\log t+r^2-t^2-\log r.\] One has \(j_t(t)=0\) and \(j_t'(r)=2r-r^{-1}\). Hence \(j_t(r)>0\) for \(0<r<t\). On \(t<r\le1\), the only possible interior critical point is a minimum, while \(j_t(1)=\log t+1-t^2<0\) for the stated range of \(t\); thus \(j_t<0\) there. Compactness gives a positive lower bound for \(j_t\) on \(\overline\Omega\), uniform over \(t\in[1/5,3/10]\).

Apply Proposition 4 to \(\phi_0\) on \(\Omega\), using a \(C^0\) tolerance smaller than both \(\varepsilon\) and half this positive lower bound. Extend the perturbation by zero on the sphere. It yields the desired integral, preserves the envelope inequalities, and preserves strictness and nonvanishing in \(\Omega\). Outside \(\Omega\) the profile is unchanged, so it is strict throughout \(r\le3/10\) and has no critical points there except \(r=0\).

At \(r=0\), the perturbation vanishes on a neighborhood and \(\,\mathrm d\phi_0=0\). The connection term in the Hessian therefore vanishes, and for every incoming metric \[H_\phi=2(\,\mathrm dx_1^2+\,\mathrm dx_2^2) \qquad\text{at }r=0.\] This form is positive semidefinite of rank two. It is positive on a two-plane, and every two-plane contains a direction on which it is strictly positive. This last observation will allow transverse phase directions to be chosen even at critical points.

Finally, shrink \(\mathcal U\) if necessary so that the metric and coordinate norms, and the coordinate volume densities on \(\overline\Omega\), are uniformly comparable. Since \(L_\phi\ge|a_\phi|\), the uniform upper bound for \(\,\mathrm dV_g/\,\mathrm dx\) proves (11). Each coordinate function on \(\overline\Omega\) extends smoothly to the sphere using a cutoff in its chart. The extensions have uniformly bounded gradients for \(g\in\mathcal U\), so their two-coordinate projections have uniform Lipschitz constants for the ambient Riemannian distance. ◻

Proof. Write \(Q=U+iD\) with \(U,D\) real symmetric. The equation \(Qz=0\) is equivalent to \[Db=Ua,\qquad Da=-Ub.\] For \(a\ne0\), these prescribed columns of a symmetric \(D\) are compatible exactly when \(U(a,a)+U(b,b)=0\). To see sufficiency explicitly, write \(a=Ae_1\), \(b=Be_2\) in an orthonormal frame. Prescribe \[De_1=-\frac BA Ue_2,\qquad De_2=\frac AB Ue_1.\] Their overlapping off-diagonal entries agree precisely when \(A^2U_{11}+B^2U_{22}=0\); symmetry prescribes the remaining entries in the first two rows, and \(D_{33}\) is free. Taking \[\alpha=\frac{H(a,a)+H(b,b)}{|a|^2+|b|^2}>0, \qquad U=H-\alpha\mathop{\mathrm{Id}}\] therefore proves the assertion.

If \(a=0\), then \(|b|=1\). Set \(U=0\) on \(\mathbb Rb\) and \(U=-M\mathop{\mathrm{Id}}\) on \(b^\perp\), with no mixed entries, and take \(D=0\). In a frame starting with \(b\), the first diagonal entry of \(H-U\) is \(H(b,b)>0\). Increasing \(M\) makes its transverse Schur complement positive definite. Hence \(H-U>0\) and \(Qb=0\).

For the continuation assertion, start with \(Qz=0\) and \(z\cdot z=-1\). For nearby \(z'\) with \(z'\cdot z'=-1\), define \[P(z')=\mathop{\mathrm{Id}}+z'(z')^T, \qquad Q'=P(z')^TQP(z').\] Then \(P(z')z'=0\), so \(Q'\) is symmetric and annihilates \(z'\). At \(z'=z\) one has \(QP(z)=P(z)Q=Q\), and hence \(Q'=Q\). Although \(P(z')\) is singular, these identities suffice: continuity preserves the strict inequality \(H'-\Re Q'>0\) for nearby \(H',z'\). No division by \(|a|\) is involved in this continuation. ◻

Proof. Use normal coordinates based on the local orthonormal frames above. The fixed smooth metric has uniform coordinate derivative bounds of every specified finite order on a common ball. Put \(z=z_\ell\) and \(Q=Q_\ell\). The quadratic initial Taylor polynomial is \[S_{\le2}(x)=\phi(p)+z\cdot x+\tfrac12x^TQx.\] The eikonal error \(E=1+g^{ij}S_iS_j\) has zero constant term because \(z\cdot z=-1\), and zero linear term because the metric’s first derivatives vanish at the normal-coordinate origin and \(Qz=0\).

Choose integers \[J=D+m+1,\qquad K=2D+3m+6, \qquad N_j=K+J-j,\qquad L_S=K+J+1.\] Construct \(S\) through degree \(L_S\) by cancelling the eikonal coefficients through degree \(L_S-1\). At degree \(d\ge2\), the new homogeneous term \(S_{d+1}\) enters only as \(2D_zS_{d+1}\); all other terms are already known. Equation (18) therefore solves each successive equation, preserving the given quadratic data. The resulting actual smooth eikonal error satisfies \[E=O(|x|^{L_S}).\] Here and below these are Taylor-vanishing statements for the actual smooth metric coefficients, not only identities between formal symbols.

Define \[T=2\mathop{\mathrm{grad}}S\cdot\nabla+\Delta_gS+2,\] where the last two terms act by multiplication. Construct \(V_j\) through degree \(N_j\), with the prescribed constant terms, successively for \(j=0,\ldots,J\). Cancel, through degree \(N_j-1\), the coefficients of \[R_0=TV_0,\qquad R_j=TV_j+\Delta_gV_{j-1}\quad(j\ge1).\] At degree \(d\) the new term of \(V_j\) again enters as \(2D_z V_{j,d+1}\), so the same right inverse applies. This finite schedule supplies all forcing terms: the largest phase degree needed is \(N_0+1=L_S\), and for \(j\ge1\) the coefficient of \(\Delta_gV_{j-1}\) of degree \(d\le N_j-1\) needs \(V_{j-1}\) only through degree \(d+2\le N_j+1=N_{j-1}\). Consequently \[R_j=O(|x|^{N_j})=O(|x|^K).\] Every operation uses finitely many metric coefficients and bounded right inverses on the compact family of phase data. Thus all the constructed coefficients have uniform bounds for these fixed orders, including as \(\delta=n^{-1/2}\) tends to zero.

The Taylor expansion of \(\Re S-\phi\) has linear part either zero or \(\delta\nu\cdot x\). Equation (17) and a common bound on third derivatives give, after decreasing the common ball, \[\Re S(x)-\phi(x)\le\delta|x|-\kappa|x|^2.\] Normal-coordinate radius equals \(\mathop{\mathrm{dist}}_g(p,x)\) on this ball, proving (20). Choose a radial cutoff supported in this ball, equal to one on a common smaller ball, and make the support small enough to lie in \(r<t_2\) for every center.

Before applying the cutoff, direct differentiation gives \[ (\Delta_g+\Lambda_n)(e^{nS}V) =e^{nS}\left(n^2EV+ \sum_{j=0}^J n^{1-j}R_j+n^{-J}\Delta_gV_J\right). \tag{23}\] For any fixed \(q\ge0\), \[ \sup_{r\ge0} r^q e^{\sqrt n r-cn r^2} \le C_{q,c}n^{-q/2}; \tag{24}\] this follows by substituting \(s=\sqrt n r\). Taylor’s Theorem bounds the derivatives of a remainder vanishing to order \(K\) by \(C|x|^{K-j}\) through order \(j\le m\). Differentiating the exponential at most \(m\) times costs at most \(Cn^m\). Thus the terms in (23) involving Taylor remainders, after at most \(m\) derivatives and then division by \(e^{n\phi}\), are bounded by \[C n^{2+m-(K-m)/2}=C n^{-D-1}.\] The last amplitude term is bounded by \(Cn^{m-J}=Cn^{-D-1}\). On the support of a derivative of the cutoff, the distance from the center has a fixed positive lower bound. For large \(n\), Equation (20) then gives a fixed negative gap between \(\Re S\) and \(\phi\), so all cutoff errors are exponentially small relative to \(e^{n\phi}\). This proves (21). Direct differentiation and (24) with \(q=0\) prove (22). Additional fixed derivative orders only require additional finite coefficient bounds. ◻

Proof. If \(f_j\to f\) uniformly, every strict opposite-sign test for \(f\) eventually remains a strict opposite-sign test for \(f_j\). Fatou’s Lemma, applied to each integral in the finite sum, proves lower semicontinuity. The two invariances follow directly from the sign of the product at its endpoints.

For the area estimate fix a cube \(P\), a direction \(i\), and write its coordinate in that direction as \(t\). Slice at levels \(t=s+kd_P\), where \(0\le s<d_P\) and \(k\in\mathbb Z\). For two consecutive levels inside \(P\), each transverse point with opposite endpoint signs has an interior zero in the intervening open slab. Hence the set of such transverse points is contained in the coordinate projection of the zeros in that slab. By the uniform Lipschitz projection bound, its two-dimensional Lebesgue measure is at most a common constant times the \(h\)-Hausdorff measure of those zeros. These sets are measurable: the zeros in an open slab are a countable union of compact sets, and so are their continuous projections. For fixed \(s\) the slabs are disjoint. Summing over \(k\), integrating over \(0\le s<d_P\), and dividing by \(d_P\) gives \[d_P^{-1}\int_{\{x\in P:x+d_Pe_i\in P\}} \mathbf1_{\{f(x)f(x+d_Pe_i)<0\}}\,dx \le C\mathcal H_h^2(\{f=0\}\cap P).\] This remains valid if the measure on the right is infinite. The open cubes are disjoint, so summing over them and over three directions proves (27). No regularity of the zero set has been used.

Finally, bounding each indicator by one gives \[\mathcal F_n(f)\le\frac{3n}{\epsilon} \sum_P\ell_P|P| \le\frac{6n}{\epsilon}\int_\Omega L(x)\,dx,\] by (25). ◻

Proof. Let \(G_n^{\rm coef}\) be the event that every coefficient has magnitude at most \(n\). There are \(O(n^3)\) coefficients, so Gaussian tails imply \(\Pr((G_n^{\rm coef})^c)\to0\). On this event, summing the individual wave estimates costs at most a factor \(Cn^4\). Together with (30) and the background residual estimate, this proves (32). In finitely many smooth local orthonormal frames, the normalized real four-component jet \[\mathcal J_n=\frac{(U_n,n^{-1}\nabla U_n)}{W_n}\] has Lipschitz constant at most \(Cn^6\) on \(G_n^{\rm coef}\). Indeed, its derivative involves only \(U_n,\nabla U_n,\nabla^2U_n\), \(W_n^{-1}\nabla W_n\), and bounded derivatives of the frames; the preceding estimates even give \(Cn^5\).

First consider points where \(r^n>n^6e^{n\phi}\). The ambient rotation field \(R=x_1\partial_{x_2}-x_2\partial_{x_1}\) is tangent to the sphere and has uniformly bounded \(g\)-length. Writing \(x_1+ix_2=re^{i\theta}\), \[w_n=r^n\cos(n\theta),\qquad Rw_n=-nr^n\sin(n\theta).\] Consequently \(|w_n|+n^{-1}|\nabla w_n|_g\ge c r^n\). The wave jet on \(G_n^{\rm coef}\) is bounded by \(Cn^4e^{n\phi}\), so domination proves (31) in this region for large \(n\).

The remaining closed set \[E_n=\{r^n\le n^6e^{n\phi}\}\] lies in \(K_1\) for large \(n\). Indeed \(\phi=\phi_0\) when \(r\ge t_1\), and \(\phi_0-\log r\) has a strictly negative maximum on that compact region. Fix \(x\in E_n\) and take a center \(p\) at distance at most \(1/n\). The three associated cutoffs are one at \(x\). Their fixed Taylor bounds and constant terms imply \[V(x)=1+O(n^{-1}),\qquad \frac{\nabla Z(x)}{nZ(x)}=z_\ell+O(n^{-1}),\qquad \frac{|Z(x)|}{W_n(x)}\ge c n^{-6},\] in a common orthonormal frame, with the initial vectors at \(p\) identified with that frame up to an \(O(n^{-1})\) change. For the last estimate, Taylor’s Theorem gives \(n|\Re S(x)-\phi(x)|\le C\) at distance \(1/n\), and \(W_n(x)\le(1+n^6)e^{n\phi(x)}\).

After factoring a complex phase and a positive modulus from each column, the covariance supplied just by these three independent coefficients is described, up to \(O(n^{-1})\) column errors, by the real and imaginary parts of \[(1,a+ib_0),\qquad (1,a+ib_1),\qquad (1,a+\delta\nu+ib'_0),\qquad \delta=n^{-1/2}.\] The resulting real \(4\) by \(6\) matrix has least singular value at least \(c\delta\). To check this, test its transpose on \((s,v)\in\mathbb R\times\mathbb R^3\). The two transverse imaginary columns control the component of \(v\) in \(\nu^\perp\), using the positive angle bound and \(L\ge1\). Subtracting the first and third real-column evaluations gives \(\delta\langle v,\nu\rangle\). The first real column then controls \(s\), since \(a=|a|\nu\) and \(|a|\) is bounded for the fixed profile. Thus \(|(s,v)|\) is bounded by \(C\delta^{-1}\) times the transpose evaluation norm. The \(O(n^{-1})\) errors preserve this estimate for large \(n\).

Multiplication of a complex coefficient by a unit complex number preserves its Gaussian law. Moreover, each factored modulus is at least \(cn^{-6}\). Therefore every unit real linear functional of \(\mathcal J_n(x)\) has standard deviation at least \(cn^{-13/2}\). Adding the other independent waves cannot decrease covariance. The deterministic jet of \(w_n\) changes only the mean. The Gaussian density in four real dimensions is consequently at most \(Cn^{26}\), and hence at most \(Cn^{28}\). In particular, \[\Pr\{ |\mathcal J_n(x)|\le2n^{-B}\} \le Cn^{28-4B},\qquad x\in E_n.\]

In each of the finitely many frame charts take a net of \(E_n\) with points in \(E_n\) and mesh at most \(n^{-B-10}\). Metric packing bounds its total cardinality by \(Cn^{3(B+10)}\). A union bound makes the probability of a small jet at any net point at most \[Cn^{28-4B+3(B+10)}=Cn^{58-B}.\] For \(B=110\) this tends to zero. On \(G_n^{\rm coef}\) the Lipschitz error from a net point to a general point is \(O(n^{-B-4})\), so the jet norm is at least \(n^{-B}\) throughout \(E_n\) on the resulting event. Its Euclidean norm is bounded above by \((|U_n|+n^{-1}|\nabla U_n|_g)/W_n\). This proves the desired lower bound everywhere. The polynomial powers used in this argument do not depend on the number of fixed Taylor or transport orders: increasing those orders changes only constants, radii, and thresholds. ◻

Proof. Write the random part of \(U_n\) as \(X_n\) and let \[\sigma_n(x)^2=\mathbb E X_n(x)^2=\sum_Z|Z(x)|^2.\] The wave at a center within distance \(1/n\) shows that \[ \sigma_n(x)\ge c e^{n\phi(x)},\qquad x\in\overline\Omega, \tag{34}\] with a fixed-stage constant \(c>0\). Since \(\min_{\overline\Omega}(\phi-\log r)>0\), the standardized mean \(w_n(x)/\sigma_n(x)\) tends uniformly to zero on \(\overline\Omega\).

Fix \(P\) and \(x\in P\) such that all three points \(y_i=x+d_Pe_i\) lie in \(P\). Use the same near-center set for every axis: retain the waves whose centers have distance at most \(n^{-1/4}\) from \(x\). A center outside this set has distance at least \(\tfrac12n^{-1/4}\) from every \(y_i\), because \(\mathop{\mathrm{dist}}_g(x,y_i)\le C d_P\le C\epsilon/n\). Equation (20) therefore bounds all these far waves, at \(x\) and all \(y_i\), by the respective \(e^{n\phi}\) factors times \(C\exp(-c\sqrt n)\) for large \(n\). After summing \(O(n^3)\) waves, their squared sums are still negligible relative to the lower variances in (34). Their covariance contributions are negligible by Cauchy–Schwarz as well.

For every retained wave, the cutoff is one at both endpoints, and uniform Taylor estimates give \[V(x)=1+o(1),\qquad \Re dS(x)=d\phi(x)+o(1),\qquad |\beta_Z|_g=L(x)+o(1),\quad \beta_Z=\Im dS(x).\] The same estimates hold in the small segments between the endpoints. Indeed their distances from the center are \(O(n^{-1/4})\), the phase and amplitude derivatives have fixed bounds, and the shifted initial real gradient differs by only \(n^{-1/2}\nu\). Every error here is uniform in centers, \(P,x\), and \(i\) for the fixed profile and orders. Taylor expansion over a segment of length \(d_P\) yields \[ \frac{Z(y_i)}{Z(x)} =\alpha_i e^{i\theta_{Zi}}(1+o(1)), \qquad \alpha_i=e^{nd_P\partial_i\phi(x)},\quad \theta_{Zi}=nd_P\beta_Z(e_i). \tag{35}\] For example, the phase Taylor remainder after multiplication by \(n\) is \(O(nd_P^2)=O(n^{-1})\). The real factor \(\alpha_i\) is common to all retained waves for the given axis. It and its reciprocal are bounded by a constant independent of the profile, since \(nd_P|\partial_i\phi(x)|\le C\epsilon\).

The covariance of \(X_n(x)\) and \(X_n(y_i)\) is \[\sum_Z\Re\bigl(\overline{Z(x)}Z(y_i)\bigr).\] Discarding the negligible far terms and using (35) in both this numerator and the variance at \(y_i\) proves that their correlation satisfies \[ \rho_i=\sum_{Z\text{ near}}\omega_Z\cos\theta_{Zi}+o(1), \qquad \omega_Z=\frac{|Z(x)|^2} {\sum_{Z'\text{ near}}|Z'(x)|^2}. \tag{36}\] The same nonnegative weights, summing to one, apply to all three axes: the common positive factor \(\alpha_i\) cancels on standardizing.

Choose constants \(c_0,C_0>0\) for the fixed equivalence of coordinate and metric covector norms, so that \[c_0|\beta|_g^2\le\sum_{i=1}^3\beta(e_i)^2 \le C_0|\beta|_g^2\] on \(\overline\Omega\) for all the incoming metrics under consideration. Choose \(\epsilon\) so small that \(2\epsilon\sqrt{C_0}\le1\). By (25) and \(L\ge1\), for large \(n\) every retained wave satisfies \[ |\theta_{Zi}|\le1,\qquad \sum_{i=1}^3\theta_{Zi}^2\ge\frac{c_0\epsilon^2}{16}. \tag{37}\] For the lower estimate it suffices to use \(|\beta_Z|_g\ge L/2\) and \(\ell_P\le2L\). These deliberately loose constants emphasize their independence from the profile.

Since \(1-\cos s\ge s^2/4\) for \(|s|\le1\), averaging Equation (37) with the common weights gives, for all sufficiently large \(n\), \[\sum_{i=1}^3(1-\rho_i)\ge\frac{c_0\epsilon^2}{128}.\] Also every \(\rho_i\ge\cos(1)+o(1)>0\). Consequently at least one axis has its correlation in a fixed compact subinterval of \((-1,1)\), bounded away from \(1\) by \(c_0\epsilon^2/384\). This reasoning still applies if a single carrier direction accounts for nearly all the variance.

For completeness, the required opposite-sign probability bound can be read directly from bivariate Gaussian densities. If the correlation belongs to a fixed compact subinterval of \((-1,1)\) and the two standardized means have absolute value at most \(1/4\), that density has a positive common minimum on the rectangle \([1,2]\times[-2,-1]\). Its integral gives a fixed positive probability of opposite signs. The standardized means here tend uniformly to zero, so there is \(p_0>0\), depending only on the fixed angular gap, such that \[\sum_{i=1}^3 \Pr\{U_n(x)U_n(x+d_Pe_i)<0\}\ge p_0\] whenever all three shifts stay in \(P\), for sufficiently large \(n\). No measurable choice of a preferred axis is needed in this inequality.

The subdivision is finite and fixed before \(n\). Thus, for large \(n\), the subset of each \(P\) on which all three shifts remain in \(P\) has volume at least \(|P|/2\). Integrating the last probability estimate and using Tonelli’s Theorem yields \[\mathbb E\mathcal F_n(U_n) \ge\frac{p_0 n}{2\epsilon}\sum_P\ell_P|P| \ge\frac{p_0 n}{2\epsilon}\int_\Omega L(x)\,dx.\] This proves Lemma 13 with a profile-independent constant. All limits used to reach its threshold were taken with the profile and finite accuracy orders fixed. ◻

Proof. Let \(G_n\) be the simultaneous event of Lemma 12, and put \(A_n=n\int_\Omega L(x)\,dx\). By Proposition 11 and Lemma 13, \[0\le\mathcal F_n(U_n)\le C_{\rm sign}A_n, \qquad \mathbb E\mathcal F_n(U_n)\ge c_{\rm sign}A_n, \qquad \Pr(G_n^c)\longrightarrow0.\] It follows without an independence assumption that \[\mathbb E\bigl[\mathcal F_n(U_n)\mathbf1_{G_n}\bigr] \ge\bigl(c_{\rm sign}-C_{\rm sign}\Pr(G_n^c)\bigr)A_n \ge\tfrac12c_{\rm sign}A_n\] for all sufficiently large \(n\). Some realization in \(G_n\) therefore has \(\mathcal F_n(U_n)\ge(c_{\rm sign}/4)A_n\). Select such a realization and set \(c_{\rm wave}=c_{\rm sign}/4\). All waves are supported in \(r<t_2\), and \(g\) is round for \(r\ge R_0\). Since \(R_0<t_2\), Equation (39) follows. ◻

Proof. The lower bound in (40) gives \[ \mathcal D\ge \tfrac12 n^{2-2B}W^2>0. \tag{45}\] Thus the formulas in (42) define smooth functions, regardless of the regularity of \(W\). Put \(f=\mathcal R/\mathcal D\). The product rule gives \[\begin{align*} \Delta_bU+\Lambda sU &=\Delta_gU-U\mathop{\mathrm{div}}_g(f\nabla U)-f|\nabla U|^2 +\Lambda U-n^2fU^2+U\mathop{\mathrm{div}}_g(f\nabla U)\\ &=\mathcal R-f\mathcal D=0. \end{align*}\] The assertion on the set where the residual vanishes follows from the same formulas.

We give a quantitative derivative estimate to make clear why an exponentially small weight causes no loss. For \(0\le j\le h+1\), the product rule and (41) yield \[|\nabla^j\mathcal D|\le C_j n^{j+10}W^2.\] For \(j\ge1\), each term in a derivative of \(\mathcal D^{-1}\) is a constant multiple of \[\mathcal D^{-p-1} (\nabla^{a_1}\mathcal D)\cdots (\nabla^{a_p}\mathcal D), \qquad a_i\ge1,\quad a_1+\cdots+a_p=j,\] with the appropriate tensor contractions. By (45), its norm is bounded by \[C_jn^{j+10p+(2B-2)(p+1)}W^{-2} =C_jn^{j+(2B+8)p+2B-2}W^{-2}.\] The same conclusion for \(j=0\) follows directly from (45). Since \(p\le j\), we may use the convenient weaker bound \[ |\nabla^j(\mathcal D^{-1})| \le C_jn^{E(j+1)}W^{-2} \qquad (0\le j\le h+1). \tag{46}\]

A derivative of \(b-1\) consists of a derivative of \(\mathcal R\), a derivative of \(U\), and a derivative of \(\mathcal D^{-1}\). The first two factors contribute one power of \(W\) each and the last contributes \(W^{-2}\). For \(j\le h\), the resulting bound is at most \[C_jn^{-D_*+8+j+E(j+1)}.\] The term \(\Lambda^{-1}\mathcal Rn^2U/\mathcal D\) has the same bound, because \(n^2/\Lambda\le1\). Finally, a derivative of order \(j\) of \(\Lambda^{-1}\mathop{\mathrm{div}}_g((\mathcal R/\mathcal D)\nabla U)\) uses derivatives of \(\mathcal R\) through order \(j+1\) and of \(U\) through order \(j+2\). Applying (46), and using \(\Lambda^{-1}\le n^{-2}\), bounds it by \[C_jn^{-D_*+E(j+2)+j+8}.\] Both displayed exponents are bounded above by \(-D_*+Q_h\). This proves (44). At no point have we differentiated \(W\); its powers cancel in every differentiated quotient. The final conclusions follow by taking \(n\) large enough to absorb the fixed constant and to make the \(C^0\) error smaller than \(1/2\). ◻

Proof. In a \(g\)-orthonormal frame the metric endomorphism is \(C=dA^{-1}\). It is positive and symmetric, and in dimension three \[\det C=\frac{d^3}{\det A}=d^2.\] Thus its positive relative volume density is \(d\), and its inverse endomorphism is \(C^{-1}=A/d\). The energy identity is consequently \[\int_M g_A(\nabla_{g_A}f,\nabla_{g_A}h)\,\,\mathrm dV_{g_A} =\int_M g(A\nabla f,\nabla h)\,\,\mathrm dV_g\] for compactly supported smooth \(f,h\). Integration by parts proves the operator identity in (48). ◻

Proof. Let \(H^a=H^a(M,g)\) denote the usual \(L^2\) Sobolev spaces. On the fixed manifold, the inverse \[L_\Lambda^{-1}=(-\Delta_g+4\Lambda)^{-1}:H^{l-2}\longrightarrow H^l\] has norm bounded independently of \(\Lambda\ge1\). Indeed, if \(\mu\ge0\) is an eigenvalue of \(-\Delta_g\), the multiplier for this map in the spectral Sobolev norms is \[\frac{1+\mu}{\mu+4\Lambda}\le1.\] The spectral and chart Sobolev norms are equivalent with constants depending on \(g\) and \(l\), not on \(\Lambda\). This equivalence follows from the compact-manifold elliptic estimate and the spectral theorem; see [7].

Write \(v=1+h\). Equation (51) becomes \[ L_\Lambda h =\mathop{\mathrm{div}}_g((b-1)\nabla h) -\Lambda\bigl(b^3(1+h)^5-s(1+h)-4h\bigr). \tag{53}\] Let \(T_n(h)\) be the result of applying \(L_\Lambda^{-1}\) to the right side. The polynomial in parentheses has constant term \(b^3-s\) and linear coefficient \[5b^3-s-4=5(b^3-1)-(s-1).\] Its remaining terms have degree at least two in \(h\).

Sobolev multiplication and (50) therefore imply, on the closed \(H^l\) ball of radius \(\delta\le1\), \[\begin{align*} \|T_n(0)\|_{H^l}&\le C\Lambda n^{-P_1},\\ \|T_n(h)-T_n(\widetilde h)\|_{H^l} &\le C\Lambda(n^{-P_1}+\delta) \|h-\widetilde h\|_{H^l}. \end{align*}\] Here the divergence term is bounded from \(H^l\) to \(H^{l-2}\) by \(C\|b-1\|_{C^l}\); the harmless factor \(\Lambda\ge1\) allows it to be included in the displayed estimate. The multiplication and embedding statements used here hold in dimension three because \(l\ge4\) and \(l>k_2+3/2\); see [20].

Set \(Q=P_2+4\) and \(\delta=n^{-Q}\). Since \(P_1=Q+4\) and \(\Lambda\le3n^2\) for \(n\ge1\), \[\|T_n(0)\|_{H^l}\le Cn^{-Q-2},\qquad \operatorname{Lip}(T_n)\le Cn^2(n^{-P_1}+n^{-Q})=o(1).\] For all sufficiently large \(n\), the map preserves this closed ball and is a contraction there. Its fixed point satisfies \(\|h\|_{H^l}\le n^{-Q}\). Sobolev embedding then gives \(\|h\|_{C^{k_2}}\le Cn^{-Q}\le n^{-P_2}\), and also \(\|h\|_{C^0}<1/2\), so \(v=1+h\) is positive.

The coefficients \(b,s\) are smooth and \(b\) is uniformly positive for large \(n\). The polynomial right side of \(\Delta_bv=\Lambda b^3v^5-\Lambda sv\) lies initially in \(H^l\). Elliptic regularity raises the regularity of \(v\) to \(H^{l+2}\); iteration gives \(v\in C^\infty\). Only smoothness, not a bound uniform in \(n\) on every derivative at once, is needed in this bootstrap; see [7]. ◻

Proof. We specify the finite orders before taking \(n\) large. Choose \[ \begin{gathered} k_2=k+2,\qquad P_2=P+k+10,\\ l\ge4\text{ even},\quad l>k_2+3/2,\qquad k_1=l,\quad P_1=P_2+8,\\ m_*=k_1+2,\qquad D_*\ge P_1+(2B+21)(k_1+2)+12. \end{gathered} \tag{59}\] These choices depend only on \(k,P,B\). We suppress the subscript \(n\) in the construction.

Apply Lemma 16 with \(h=k_1\) to obtain positive smooth coefficients \(b,s\) satisfying (43) and (50). Since \(g\) is round and \(U=w_n\) on \(\{r\ge t_2\}\), their residual vanishes there, and \(b=s=1\) there. Apply Lemma 18 to obtain a positive smooth solution \(v\) of (51) satisfying (52).

Choose numbers \(t_2<a<b_0<t_3\), and fix a smooth cutoff \(\chi\) with \(0\le\chi\le1\), which is one on \(\{r\le a\}\) and zero on \(\{r\ge b_0\}\). Such a cutoff is smooth on the sphere because it is constant near \(r=0\). Set \[ \begin{aligned} \widetilde v&=1+\chi(v-1),\qquad u=U/\widetilde v,\\ q&=b\widetilde v^2,\qquad d=s\widetilde v^2+\Lambda^{-1}\widetilde v\Delta_b\widetilde v. \end{aligned} \tag{60}\] For all sufficiently large \(n\), \(\widetilde v\) is positive and \(\|\widetilde v-1\|_{C^{k+2}}\le Cn^{-P_2}\). Applying the division identity (49) with \(v=\widetilde v\) gives \[ \mathop{\mathrm{div}}_g(q\nabla u)+\Lambda d u=0. \tag{61}\]

The coefficient estimates and the two extra derivatives included in \(k_2=k+2\) give \[ \|q-1\|_{C^k}+\|d-1\|_{C^k}\le Cn^{-P_2}. \tag{62}\] For the term \(\Delta_b\widetilde v\), one uses \(\Delta_b\widetilde v=b\Delta_g\widetilde v+ g(\nabla b,\nabla\widetilde v)\), so the required derivatives of \(b\) are available because \(k_1\ge k+1\). In particular \(q,d>0\) for large \(n\).

On the open set \(\{r<a\}\), the cutoff is identically one, and (51) gives \[d=sv^2+\Lambda^{-1}v\Delta_bv=b^3v^6=q^3.\] Where \(r\ge b_0\), we have \(\widetilde v=1\) and \(b=s=1\), so \(q=d=1\). Consequently the defect \(d-q^3\) is supported in the compact annulus \(\{a\le r\le b_0\}\), strictly inside \(\{t_2<r<t_3\}\). Without this defect, the choice \(A=q\mathop{\mathrm{Id}}\) would already have the required determinant \(d\). We correct that determinant on the two-plane perpendicular to \(\nabla u\).

We first justify that this plane and its derivatives are controlled throughout the transition annulus. The round gradient of \(w_n\) satisfies \[ |\nabla w_n|_{g_*}^2 =n^2(r^{2n-2}-w_n^2) \ge n^2r^{2n-2}(1-r^2). \tag{63}\] To verify the equality, the ambient Euclidean gradient has squared norm \(n^2r^{2n-2}\), while its normal component on the unit sphere is \(nw_n\), by homogeneity. Since \(U=w_n\) here and \(\widetilde v\) is close to one in \(C^1\), \[|\nabla u| \ge \widetilde v^{-1}nr^{n-1}\sqrt{1-r^2} -\widetilde v^{-2}r^n|\nabla\widetilde v| \ge c n r^n>0 \quad(t_2<r<t_3)\] for large \(n\).

For the derivative estimates, write locally \(x_1+ix_2=re^{i\theta}\) and set \[\alpha_n=\frac{\cos(n\theta)}r\nabla r -\sin(n\theta)\nabla\theta.\] The displayed expression is globally defined on the annulus: the trigonometric functions are single-valued and \(\,\mathrm d\theta=(-x_2\,\mathrm dx_1+x_1\,\mathrm dx_2)/r^2\) is a smooth one-form there. On the fixed annulus, \[|\alpha_n|^2=r^{-2}-\cos^2(n\theta)\ge t_3^{-2}-1>0, \qquad |\nabla^j\alpha_n|\le C_j n^j.\] Moreover \[\nabla u=nr^n t_n,\qquad t_n=\widetilde v^{-1}\alpha_n -n^{-1}\cos(n\theta)\widetilde v^{-2}\nabla\widetilde v.\] It follows that \(|t_n|\ge c>0\) and \(|\nabla^jt_n|\le C_jn^j\) for \(0\le j\le k\). The orthogonal projection \[P_\perp=\mathop{\mathrm{Id}}-\frac{t_n\otimes t_n^\flat}{|t_n|^2}\] is therefore smooth and satisfies \[ |\nabla^jP_\perp|\le C_jn^j\qquad(0\le j\le k). \tag{64}\] Here \(t_n^\flat=g(t_n,\cdot)\), so \((t_n\otimes t_n^\flat)X=g(t_n,X)t_n\). To obtain (64), differentiate the quotient defining \(P_\perp\); its denominator is uniformly bounded away from zero, and the total derivative order of the numerator factors in every term is at most \(j\). This estimate shows explicitly that only \(k\) powers of \(n\) are lost in the required projection norms.

Define a positive symmetric endomorphism by \[ A=q\left[\mathop{\mathrm{Id}}+\left(\sqrt{d/q^3}-1\right)P_\perp\right] \quad\hbox{on }\{t_2<r<t_3\},\qquad A=q\mathop{\mathrm{Id}}\quad\hbox{elsewhere}. \tag{65}\] The coefficient of \(P_\perp\) vanishes on neighborhoods of both boundary components of this annulus, because \(d=q^3\) there. Thus the two formulas agree on open collars and define a globally smooth endomorphism. Its eigenvalue along \(\nabla u\) is \(q\), and its two transverse eigenvalues are both \(q\sqrt{d/q^3}\). Hence, throughout the annulus, \[A\nabla u=q\nabla u,\qquad \det A=d.\] These identities also hold outside it, where \(A=q\mathop{\mathrm{Id}}\) and \(d=q^3\).

By (62), smooth composition gives \(\|\sqrt{d/q^3}-1\|_{C^k}\le Cn^{-P_2}\). The product rule and (64) consequently give \[\|A-\mathop{\mathrm{Id}}\|_{C^k}\le Cn^{k-P_2}=Cn^{-P-10}.\] Define \(\widehat g=g_A\) by (47). The map \((d,A)\mapsto dA^{-1}\) is smooth near \((1,\mathop{\mathrm{Id}})\), so (62) and this last estimate imply \[\|\widehat g-g\|_{C^k}\le Cn^{-P-10}\le n^{-P}\] for all sufficiently large \(n\). Combining Lemma 17 with (61) proves \(\Delta_{\widehat g}u=-\Lambda u\). On \(\{r\ge t_3\}\), \(A=\mathop{\mathrm{Id}}\), \(\widetilde v=1\), and \(U=w_n\), which proves (58). The lower bound (54) excludes \(U\equiv0\), and division by the positive function \(\widetilde v\) preserves all pointwise signs and zeros. ◻

Proof. Fix \(0<r_0<r_1<R<1/2\) and a nonzero nonnegative function \(s\in C_c^\infty((r_0,r_1))\). In the disk of radius \(R\) about each point of \(\mathbb Z^2\), define \[q(r)=-\int_r^R s(t)\,dt,\] and set \(q=0\) off the disks. The resulting function on \(\mathbb R^2\) is smooth and \(\mathbb Z^2\)-periodic: it is a negative constant near each disk center and zero near each disk boundary. Where its derivatives are nonzero, let \(n,t\) be the radial and tangential Euclidean unit vectors. Then \[ Dq=s\,n^\flat,\qquad D^2q=s'\,n^\flat\otimes n^\flat +\frac{s}{r}\,t^\flat\otimes t^\flat. \tag{80}\] Outside those annuli all derivatives in (80) vanish. Fix also \(0\le\chi\in C_c^\infty((-1,1)^4)\), equal to one on a smaller concentric box. Nonnegativity and bounded second derivatives give \[ |s'|\le C\sqrt{s},\qquad |D\chi|\le C\sqrt{\chi}. \tag{81}\] For example, for a nonnegative one-variable function \(v\) with \(\|v''\|_\infty\le L\), Taylor’s inequality at displacement \(-v'/L\) gives \(|v'|^2\le2Lv\); applying this along lines proves the second estimate as well. Let \[K_q=\sup s(s')_-,\qquad (s')_-=\max\{-s',0\},\] and fix once and for all \(0<\delta\le1\) so small that \[ 2K_q\delta^2\le\frac1{8R}. \tag{82}\] This choice is independent of \(f\), \(h\), and all subsequent amplifications.

Proof. The indicated compact set lies inside the coordinate cube \(P\), so nested coordinate boxes with these strict inclusions exist. Choose \(\zeta\in C_c^\infty(U_1)\) equal to one on a neighborhood of \(\overline U_0\). For a sufficiently small fixed \(\tau>0\), \[f^{(1)}=f_0+\tau(2\zeta-1)\] is still strict on a neighborhood of \(\overline P\), by the \(C^2\)-openness of (76). Its gaps relative to \(f_0\) are exactly \(\tau\) on \(\overline U_0\) and \(-\tau\) outside \(U_1\). Apply Lemma 26 finitely many times in \(D\), with total sup-norm change less than \(\tau/4\), to make \(\int_D|\nabla_hf|_h\,dx>L_*\). The resulting \(f\) is unchanged outside a compact subset of \(D\) and remains strict everywhere in question. Taking \(\sigma=\tau/2\) gives (93); the inequality \(B\ge|\nabla_hf|_h\) gives (94). ◻

Proof. We give a finite Taylor construction using the complex-phase WKB method; compare [4]. Fix a center and an index and temporarily suppress them. The eikonal equation is \[ \langle d\Phi,d\Phi\rangle_h=-2, \tag{101}\] with the complex bilinear extension of the metric pairing. Its constant term holds because \(a\perp b\) and \(|b|_h^2=2+|a|_h^2\). The degree-one condition is \[(\mathop{\mathrm{Hess}}_h\Phi)_y(a+i b,\cdot)=0.\] To prescribe the real Hessian, put \(U=H_y-c_{yj}h_y\). A real symmetric form \(V\) must then satisfy \[ V(b,\cdot)=U(a,\cdot),\qquad V(a,\cdot)=-U(b,\cdot). \tag{102}\] The two rows of a symmetric form have precisely one compatibility condition, namely \(U(a,a)+U(b,b)=0\). Our choice of \(c_{yj}\) gives this identity. Define the remaining block of \(V\) on \(\{a,b\}^\perp\) to be zero. Since \(a,b\) are orthogonal with lengths bounded above and away from zero, this constructs \(V\) with uniformly bounded coefficients. Thus \(U+iV\) is the required covariant Hessian; the connection converts it to the ordinary quadratic Taylor coefficients of \(\Phi\).

For completeness, the higher homogeneous equations are also explicitly soluble. In coordinates write \(v=a+i b\) and \(D_v=v\cdot\partial\). For a complex homogeneous polynomial \(F_q\) of degree \(q\), choose a complex linear function \(\zeta\) with \(D_v\zeta=1\). For example, \[\zeta(z)=\frac{\sum_i\overline{v_i}z_i}{\sum_i|v_i|^2}.\] Then \[ \mathcal R_qF_q =\sum_{r=0}^q\frac{(-1)^r\zeta^{r+1}}{(r+1)!}D_v^rF_q, \qquad D_v\mathcal R_qF_q=F_q. \tag{103}\] The identity follows by telescoping after differentiation. This is a right inverse from degree \(q\) to degree \(q+1\), with bounded norm for each fixed degree, uniformly in our data. For \(q\ge2\), the degree-\(q\) eikonal equation contains the new degree-\((q+1)\) phase coefficient only as \(2D_v\) applied to it. All other coefficients in that equation have already been determined. Hence we can match (101) through any prescribed finite Taylor degree.

The transport operator for the eigenvalue \(\Lambda_k=2k^2+k\) is \[\mathcal T=2\mathop{\mathrm{grad}}_h\Phi\cdot\partial+\Delta_h\Phi+1.\] Indeed, without the cutoff, \[ e^{-k\Phi}(\Delta_h+\Lambda_k)(e^{k\Phi}A_k) =k^2E A_k+k\mathcal T A_k+\Delta_h A_k, \qquad E=\langle d\Phi,d\Phi\rangle_h+2. \tag{104}\] Here is one finite choice of all the truncation orders. Set \[M=J+\ell+1,\qquad d=2J+3\ell+6,\qquad n_m=d+2(M-m)\ (0\le m\le M),\qquad N=n_0+1.\] Construct \(\Phi\) of degree \(N\) so that \(E\) vanishes to order \(N\) at the center. With this phase fixed, successively construct polynomials \(A^{(m)}\) of degree \(n_m\), with \(A^{(0)}(y)=1\) and \(A^{(m)}(y)=0\) for \(m\ge1\), satisfying \[\begin{align*} \mathcal T A^{(0)}&=O(|x-y|^{n_0}),\\ \mathcal T A^{(m)}+\Delta_h A^{(m-1)} &=O(|x-y|^{n_m})\qquad(1\le m\le M). \tag{105}\end{align*}\] To see that this is possible, solve the homogeneous equations of degrees \(q=0,\ldots,n_m-1\) in order. The new coefficient of \(A^{(m)}\) has degree \(q+1\) and enters as \(2D_v\) applied to it, so (103) applies. The source \(\Delta_h A^{(m-1)}\) uses coefficients of the preceding amplitude of degree at most \(q+2\le n_m+1<n_{m-1}\). Those coefficients have already been retained. The tapering degrees thus ensure that no later truncation changes an earlier equation. Only finitely many jets of the smooth coefficients of \(h\) are used. This argument is algebraic and makes no convergence assertion about an infinite formal series.

Set \(A_k=\sum_{m=0}^M k^{-m}A^{(m)}\). All its fixed-order derivatives are bounded uniformly for \(k\ge1\), and it has the stated value and first-order estimate at the center. Every Taylor remainder in (105) vanishes to order at least \(d\), and the eikonal remainder vanishes to still higher order. Uniform bounds for their differentiated remainders follow from ordinary smooth Taylor estimates and the finite bounded constructions above.

The function \(\Re\Phi-f\) has zero value and differential at the center. Its ordinary Hessian there is \(-c_{yj}h_y\): the connection terms cancel because its differential is zero. Uniform Taylor bounds therefore give (98) after choosing a sufficiently small common radius. They also give the expansion, used below, \[ d(\Re\Phi_{yj}-f)(x) =-c_{yj}h_y(x-y,\cdot)+O(|x-y|^2). \tag{106}\] Choose \(\xi\) within this radius. Differentiating its product with the exponential and amplitude proves (99).

Finally, for \(q\ge0\) the elementary Gaussian bound \[\sup_{r\ge0}r^q e^{-ckr^2}\le C_q k^{-q/2}\] converts Taylor vanishing into powers of \(k^{-1}\). Through \(\ell\) derivatives, the terms in (104) that contain matched errors are bounded in the \(e^{kf}\) scale by \[C k^{2+\ell-(d-\ell)/2}\le Ck^{-J-1}.\] The last uncancelled term is \(k^{-M}\Delta_h A^{(M)}\); its corresponding bound is \(Ck^{\ell-M}=Ck^{-J-1}\). Derivatives of \(\xi\) are supported at a fixed positive distance from the center and contribute an exponentially small remainder. This proves (100). Increasing the finite Taylor orders changes constants and radii, but introduces no new power of \(k\) in (99). ◻

Proof. Summing (99) over \(O(k^4)\) packets and using the coefficient bound \(k\) proves (108), including the fixed cutoff and the elementary bounds for \(u_k^0\). The summed packet residual in (109) costs at most five powers of \(k\).

There are two other residual terms. The background residual is \[(\Delta_h+\Lambda_k)u_k^0=(\Delta_h-\Delta_{g_0})u_k^0,\] supported in \(\mathop{\mathrm{supp}}(h-g_0)\subset U_0\). Each fixed-order derivative is bounded by a fixed power of \(k\) times \(e^{kf_0}\). The first gap of Lemma 27, \(f>f_0+\sigma\) on \(\overline U_0\), makes this exponentially small relative to \(W\). The commutator terms \([\Delta_h,\omega]p_{yj,k}\) and their derivatives are supported outside \(U_1\). There the opposite gap \(f<f_0-\sigma\) makes their polynomial bounds times \(e^{kf}\) exponentially small relative to \(W\). These are all the residual terms. Their supports, and the exact background equation off \(U_0\), prove the support assertion. ◻

Proof. Put \(\mathcal V_k=W^{-1}(u,k^{-1}du)\). First consider a point where \[ e^{kf(x)}<k^{-8}W(x). \tag{111}\] On \(\mathcal E_k\) the random contribution to the unscaled value and normalized gradient is at most \(Ck^5e^{kf(x)}\le Ck^{-3}W(x)\). The background gradient bound in Lemma 25 gives \[|(u_k^0,k^{-1}du_k^0)|\ge c e^{kf_0(x)} \ge c(1-k^{-8})W(x).\] Thus \(|\mathcal V_k(x)|\) is bounded below by a fixed positive number throughout this regime, for all large \(k\).

It remains to estimate small-ball probabilities at a fixed point \(x\in\overline U_2\) with \[ e^{kf(x)}\ge k^{-8}W(x). \tag{112}\] The gap \(f<f_0-\sigma\) outside \(U_1\) implies that, for large \(k\), such a point lies where \(\omega=1\) on a neighborhood. Thus no derivative of \(\omega\) enters this calculation. Choose a grid center \(y\) within \(Ck^{-1}\) of \(x\), and another center \(y'\) within \(Ck^{-1}\) of \[x+k^{-1/2}\nu,\qquad \nu=\frac{a(x)}{|a(x)|_h},\] viewing the tangent vector in the fixed coordinates. The compact inclusion \(\overline U_2\subset U_3\) ensures that both centers belong to \(Y_k\). They are distinct for all large \(k\). The cutoff \(\xi\) is one at \(x\) for each selected packet.

For the three packets at \(y\) and the packet with index \(1\) at \(y'\), the positive numbers \[\beta_{zj}=e^{-kf(x)}|p_{zj,k}(x)|\qquad(z=y\text{ or }y')\] are bounded above and away from zero. This follows from both sides of (98), since \(|x-z|=O(k^{-1/2})\), and from \(A_{zj,k}(x)=1+O(k^{-1/2})\).

At this fixed evaluation point, rotate the two Gaussian coordinates of each selected packet by the argument of \(p_{zj,k}(x)\). This is a fixed orthogonal change of its two real coordinates, preserving their independent standard Gaussian distribution. After factoring out \(\beta_{zj}\), the two columns of the evaluation map in the \(e^{kf(x)}\) scale are, up to changing the sign of the second coordinate, \[ \begin{pmatrix}1\\ d\Re\Phi_{zj}(x)+k^{-1}\Re(A_{zj,k}^{-1}dA_{zj,k})(x) \end{pmatrix},\qquad \begin{pmatrix}0\\ d\Im\Phi_{zj}(x)+k^{-1}\Im(A_{zj,k}^{-1}dA_{zj,k})(x) \end{pmatrix}. \tag{113}\] The rotation is held fixed in computing these derivatives. The logarithmic amplitude derivatives appearing here are uniformly bounded at the selected points.

Select the first column for the packet \((y,1)\), the second columns for \((y,1),(y,2),(y,3)\), and the first column for \((y',1)\). These are five independent real Gaussian coordinates: the first two come from the same orthogonally rotated Gaussian pair, and the other three come from different pairs. The near-center columns have the forms \[(1,df(x)+O(k^{-1})),\qquad (0,b_j^\flat(y)+O(k^{-1}))\quad(1\le j\le3).\] For the displaced first column, (106) gives \[\begin{align*} d\Re\Phi_{y'1}(x)-df(x) &=-c_{y'1}h_{y'}(x-y',\cdot)+O(k^{-1})\\ &=k^{-1/2}c_{y'1}h_x(\nu,\cdot)+O(k^{-1}). \end{align*}\] Subtracting the near first column from this column leaves zero in the value component. Multiplying the resulting column by \(k^{1/2}\) leaves the derivative covector \(c_{y'1}h_x(\nu,\cdot)+O(k^{-1/2})\). The other three derivative covectors are uniformly independent in \(a(y)^\perp\). Since \(|x-y|=O(k^{-1})\), they remain uniformly independent together with \(h_x(\nu,\cdot)\). This comparison uses smoothness of \(a,h\), and does not compare the triples chosen at different centers. The determinant of these five amplitude-factored columns is therefore at least \(ck^{-1/2}\) in absolute value. Restoring the four bounded positive packet amplitudes, with the amplitude of \((y,1)\) used twice, preserves this lower bound.

Changing the normalization from \(e^{kf(x)}\) to \(W(x)\) multiplies each of the five columns by \(e^{kf(x)}/W(x)\ge k^{-8}\). Their determinant in the \(W\) scale is consequently at least \(ck^{-40-1/2}\). Condition on all the unused rotated Gaussian coordinates. The selected five are still independent standard real Gaussians; the conditioned coordinates and the deterministic background give an arbitrary translation. Change of variables in their bounded Gaussian density now yields, unconditionally as well, \[ \mathbb P\{\,|\mathcal V_k(x)|<r\,\} \le C k^{41}r^5\qquad(r>0) \tag{114}\] at every point in (112). This estimate has not conditioned on the coefficient event \(\mathcal E_k\).

Take a net of mesh at most \(k^{-208}\) in the compact coordinate set \(\overline U_2\). It has \(O(k^{832})\) points. At its high-regime points apply (114) with \(r=2k^{-200}\). The probability of any such small value is at most \[C k^{832+41-1000}=Ck^{-127}.\] At its low-regime points the preceding deterministic bound applies on \(\mathcal E_k\). Thus, outside an event of probability tending to zero, all net values have norm at least \(2k^{-200}\) and \(\mathcal E_k\) holds.

To pass from the net to the whole set, note that \[W^{-1}|\partial^\alpha W|\le C_\alpha k^{|\alpha|},\qquad |d\log W|\le Ck.\] Together with (108) through order two, these give \(\|d\mathcal V_k\|\le Ck^6\) on \(\mathcal E_k\). The net interpolation error is at most \(Ck^{6-208}=Ck^{-202}\), which is smaller than \(k^{-200}\) for all large \(k\). This proves (110), even when a point and its chosen net point belong to different regimes. Intersecting the coefficient and net events used only a union bound, so no assertion about a truncated Gaussian density is needed. ◻

Proof. Choose a fixed smooth function \(\gamma_0:\mathbb R\to[0,1]\), equal to one on \([-1/4,1/4]\) and supported in \((-1/2,1/2)\), and set on \(U_2\) \[\gamma=\gamma_0(k^{200}u/W).\] On the support of \(\gamma\), [corr:noncancellation] gives \(|du|_h\ge c k^{-199}W\). Wherever \(1-\gamma\ne0\), \(|u|\ge\tfrac14 k^{-200}W\). Define on \(U_2\) \[\begin{align*} F&=-u\gamma R|du|_h^{-2}h^{-1},\tag{129}\\ \vartheta&=\Lambda_k^{-1}\left( -(1-\gamma)\frac R u +\mathop{\mathrm{div}}_h\bigl(\gamma R|du|_h^{-2}\mathop{\mathrm{grad}}_hu\bigr) \right). \tag{130}\end{align*}\] Each quotient is used only where its denominator is nonzero. Extend the cutoff quotient by zero on the unused part of its domain. This is smooth: near a zero of \(du\), the first cutoff is identically zero, and near a zero of \(u\), the second cutoff is identically zero. The support condition on \(R\) then permits smooth extension of both corrections by zero to \(M\), including the divergence term.

Put \(q=\gamma R|du|_h^{-2}\). The product rule gives the exact identities \[\begin{align*} \mathop{\mathrm{div}}_h(Fdu)&=-\gamma R-u\mathop{\mathrm{div}}_h(q\mathop{\mathrm{grad}}_hu),\\ \Lambda_k\vartheta u&=-(1-\gamma)R +u\mathop{\mathrm{div}}_h(q\mathop{\mathrm{grad}}_hu). \end{align*}\] Adding these identities to the definition of \(R\) proves (127), also at the zeros of \(u\), by smoothness.

For the estimates, use normalized quantities \[a_0=u/W,\qquad \beta=du/W,\qquad r_0=R/W.\] All exponentially varying factors cancel from the formulas: \[\begin{align*} F&=-a_0\gamma r_0|\beta|_h^{-2}h^{-1},\tag{131}\\ q\mathop{\mathrm{grad}}_hu&=\gamma r_0|\beta|_h^{-2}\beta^{\sharp_h}, \qquad (1-\gamma)R/u=(1-\gamma)r_0/a_0. \tag{132}\end{align*}\] Since \(f,f_0\) are fixed, differentiation of their exponentials and the reciprocal of their positive sum gives \[|\partial^jW|\le C_j k^jW, \qquad |\partial^j(W^{-1})|\le C_j k^jW^{-1}.\] Consequently, for every required order \(j\), \[ |\partial^j a_0|\le C_jk^{j+5},\qquad |\partial^j\beta|\le C_jk^{j+6},\qquad |\partial^j r_0|\le C_jk^{j-J}. \tag{133}\] Here and below a bound on \(\partial^j\) means the bound for every multi-index of that length. The chain and quotient rules give, on the respective cutoff domains, \[\begin{align*} |\partial^j\gamma|&\le C_jk^{206j},\quad j\ge1,\\ |\partial^j(a_0^{-1})|&\le C_jk^{200+206j},\\ |\partial^j(|\beta|_h^{-2})|&\le C_jk^{398+411j}. \end{align*}\] For example, \(|\beta|_h^2\ge c k^{-398}\) on the first domain, and \(|\partial^j(|\beta|_h^2)|\le C_jk^{j+12}\). Each term in its differentiated reciprocal has at most \(j\) differentiated factors, which proves the last displayed bound. These deliberately generous powers also bound derivatives at cutoff interfaces. Substitution into (131)–(132), with one extra derivative for the divergence in (130), proves (128). In particular, the loss depends only on the requested derivative order and the fixed exponent in [corr:noncancellation], and does not grow when \(J\) is increased. ◻

Proof. Use the real Sobolev spaces of the fixed metric \(h\). The standard elliptic spectral and Sobolev theorems give the eigenbasis for \(-\Delta_h\), the equivalence of the usual even-order Sobolev norms with the norms defined by powers of \(1-\Delta_h\), and the algebra property of \(H^s\) for \(s>2\) [21, 22]. If \(\mu\ge0\) is a spectral value of \(-\Delta_h\), the inverse of \(L_\Lambda=\Delta_h-2\Lambda\) has multiplier \(-1/(\mu+2\Lambda)\). In the spectral Sobolev norms, therefore, \[\begin{align*} \|L_\Lambda^{-1}\|_{H^s\to H^s} &\le (2\Lambda)^{-1},\tag{140}\\ \|L_\Lambda^{-1}\|_{H^{s-2}\to H^s} &\le\sup_{\mu\ge0}\frac{1+\mu}{\mu+2\Lambda} \le1. \tag{141}\end{align*}\] Changing to the fixed coordinate Sobolev norms costs constants independent of \(\Lambda\).

Writing \(Z=1+Y\), the required fixed-point equation is \[ Y=L_\Lambda^{-1}\left[ (\Delta_h-\Delta_{h_1})Y+V_0(1+Y) +\Lambda(3Y^2+Y^3)\right]. \tag{142}\] The coefficient formula for a Laplacian and the usual integer-order product rule show \[\|\Delta_h-\Delta_{h_1}\|_{H^s\to H^{s-2}} \le C_s\|h_1-h\|_{C^s}.\] Estimate this term using (141). Estimate the remaining terms using the algebra property and (140). On the closed \(H^s\)-ball of radius \(r\), the Lipschitz constant of the right side of (142) is at most \[C_s\|h_1-h\|_{C^s} +C_s\Lambda^{-1}\|V_0\|_{H^s}+C_s(r+r^2).\] Choose a fixed small \(r\) so that the last term is at most \(1/4\) and Sobolev embedding makes \(1+Y\ge1/2\) on this ball. Next choose \(\delta\) so that the displayed Lipschitz constant is at most \(1/2\) and the image of zero has norm at most \(r/2\). The map is then a contraction of the ball into itself. Its fixed point obeys \[\|Y\|_{H^s} \le\tfrac12\|Y\|_{H^s} +C_s\Lambda^{-1}\|V_0\|_{H^s},\] which proves (139). Finally the equation for \(Z\), the smoothness of its coefficients, and elliptic regularity bootstrap \(Z\) from \(H^s\) to \(H^{s+2}\), and then to all higher orders. The resulting real smooth solution remains positive. ◻

Proof. Put \(\phi=\theta+z\). Since \(u_k^0=e^{kf_0}\cos(k\phi)\), the normalized covector on \(\mathcal C\) is exactly \[ \alpha:=\frac{dt}{ke^{kf_0}} =\widehat Z^{-1} \bigl(\cos(k\phi)\,df_0-\sin(k\phi)\,d\phi\bigr) -k^{-1}\cos(k\phi)\widehat Z^{-2}d\widehat Z. \tag{148}\] On \(\overline U_4\), the background covectors \(df_0\) and \(d\phi\) are orthogonal in \(g_0\), with lengths bounded below. It follows that \(|\alpha|_{h_1}\ge c>0\) for all sufficiently large \(k\). In particular, \[ |dt|_{h_1}\ge ck e^{kf_0}\quad\hbox{on }\mathcal C. \tag{149}\] Let \[\Pi=I-\frac{\alpha^{\sharp_{h_1}}\otimes\alpha} {|\alpha|_{h_1}^2}\] be the \(h_1\)-orthogonal projection of the tangent bundle onto \(\ker dt\) on \(\mathcal C\). It has rank three and annihilates \(\mathop{\mathrm{grad}}_{h_1}t\).

Define the positive scalar \[c_* =\left(\frac{\rho^2}{\widehat Z^8}\right)^{1/3}.\] By (145), \(c_*-1\) has compact support in \(\mathcal C\). Consequently \((c_*-1)\Pi\) extends smoothly by zero to \(M\). Define a positive symmetric contravariant tensor and an ordinary metric by \[ A'=\widehat Z^2\bigl(I+(c_*-1)\Pi\bigr)h_1^{-1}, \qquad g_*^{-1}=\rho^{-1}A'. \tag{150}\] Here the product containing \(\Pi\) always means this smooth zero extension. In the normal direction to \(\ker dt\), the endomorphism \(A'h_1\) has eigenvalue \(\widehat Z^2\); in the three tangent directions its eigenvalue is \(\widehat Z^2c_*\). Hence \[ A'dt=\widehat Z^2h_1^{-1}dt, \qquad \det(A'h_1)=\widehat Z^8c_*^3=\rho^2. \tag{151}\] Off the collar the same identities follow from \(c_*=1\) and \(\rho=\widehat Z^4\). Dimension four now gives \[\det(g_*^{-1}h_1)=\rho^{-4}\det(A'h_1)=\rho^{-2}, \qquad d\mathrm{vol}_{g_*}=\rho\,d\mathrm{vol}_{h_1}.\] It follows from (144) and (151) that \[\Delta_{g_*}t =\rho^{-1}\mathop{\mathrm{div}}_{h_1}(A'dt)=-\Lambda_kt.\] All changes are compactly supported in \(U_4\).

It remains to verify the estimates without losing an exponential factor from \(dt\). Under (146), \(\widehat Z\ge1/2\) and its derivatives through order \(p+2\) are bounded independently of \(k\). Formula (148) gives \[|\partial^j\alpha|\le C_jk^j, \qquad |\partial^j\Pi|\le C_jk^j, \qquad 0\le j\le p.\] The second estimate follows from the first, the fixed positive lower bound on \(|\alpha|\), and the quotient rule. These powers are independent of the accuracy requested in the residual. On the region where \(\chi\equiv1\), use \(\rho=Z^4\). In the transition region use (143) with the fixed coefficients \(h_1=g_0\), \(V_0=0\), and outside \(\mathop{\mathrm{supp}}\chi\) use \(\rho=1\). Since \(\Lambda_k\) is comparable to \(k^2\), (146) therefore gives \[\|\rho-1\|_{C^p}+\|c_*-1\|_{C^p}=O(k^{-N-2}).\] The product rule therefore yields \[\|(c_*-1)\Pi\|_{C^p}=O(k^{p-N-2}).\] For the covariant metric, use the exact inverse supplied by \(\Pi^2=\Pi\): \[g_*=\rho\widehat Z^{-2}h_1 \bigl(I+(c_*^{-1}-1)\Pi\bigr).\] The scalar \(c_*^{-1}-1\), like \(c_*-1\), has \(C^p\) norm \(O(k^{-N-2})\). The product rule in this formula, together with \(\|h_1-h\|_{C^p}=O(k^{-N})\), proves (147) for every \(N>0\), including when its second term grows with \(k\). ◻

Proof of Lemma 35. Here is one explicit order of choices. Given \(p,L\), choose an even integer \(s>p+4\), and set \[\begin{gathered} N=\lceil L\rceil+p+4,\qquad r=s+2,\qquad \ell=s+3,\\ J=E_{s+2}+N,\qquad E_{s+2}=1000(s+4). \end{gathered}\] Lemma 36 uses \(u\) through order \(s+4=\ell+1\) and \(R\) through order \(\ell\), and gives \[\|F\|_{C^{s+2}}+\|\vartheta\|_{C^{s+2}}=O(k^{-N}).\] Equation (136) then gives \[\|h_1-h\|_{C^{s+2}}+\|w-1\|_{C^{s+2}} +\|V_0\|_{H^s}=O(k^{-N}).\] For large \(k\), Lemma 37 applies with \(\Lambda=\Lambda_k\). Its estimate and the embedding \(H^s\hookrightarrow C^{p+2}\), valid because \(s>p+4\) in dimension four, give \[\|Z-1\|_{C^{p+2}} \le C\|Z-1\|_{H^s}=O(k^{-N-2}).\] The hypotheses of Lemma 38 now hold. Since \(N\ge L+p+4\), its conclusion implies \(\|g_*-h\|_{C^p}=O(k^{-L})\). The eigenfunction is exactly \[t=\frac{w^{1/2}}{\widehat Z}\,u,\] with a smooth positive multiplier equal to one off a compact subset of \(U_4\). This proves every assertion.

All derivative orders and powers have been fixed before the wave construction is asked to provide [corr:derivatives]–[corr:residual]. The constants and the eventual lower threshold for \(k\) may depend on that finite construction. Increasing its Taylor orders changes those constants, but neither the fixed noncancellation exponent nor the polynomial losses above. Thus only finitely many accuracy requirements are used at each realization step. ◻