Proof. In dimension two the normalized Dirichlet form of \(h^0(w)\) is \[q(f,g)=\int_{S^2}\langle\nabla_0 f,\nabla_0g\rangle\,d\mu,\] independent of \(w\). Its mass form is \(m_w(f,g)=\int f g\,d\mu_w\). Differentiating the latter gives \[\dot m_w(f,g)= 2\int_{S^2}fg \left(\dot w-\int_{S^2}\dot w\,d\mu_w\right)d\mu_w.\] For an \(m_w\)-orthonormal basis \(u_1,\ldots,u_r\) of a repeated positive eigenspace, differentiation of the generalized eigenvalue equation \(q=\lambda m_w\) therefore gives the symmetric cluster matrix \[ \dot M_{ij}= -2\lambda\int_{S^2}u_i u_j \left(\dot w-\int_{S^2}\dot w\,d\mu_w\right)d\mu_w . \tag{7}\] Terms arising from a change of orthonormal frame cancel at a scalar cluster matrix. The mean term in (7) is scalar, so the two traceless entries on the pair \(u,v\) are represented, up to fixed nonzero factors, by \[ U_1=u^2-v^2,\qquad U_2=2uv. \tag{8}\] Both have mean zero for \(d\mu_w\).

The functions \(U_1,U_2\) are linearly independent. Indeed, a nonzero linear combination of them is a nonzero traceless quadratic form in \((u,v)\); its zero set in \(\mathbb R^2\) is the union of two lines. If that quadratic form vanished at every point of the sphere, choose a point where \((u,v)\ne(0,0)\) and a connected neighborhood on which this remains true. Continuity then puts \((u,v)\) in just one of the two lines on a smaller neighborhood. A fixed nonzero linear combination of \(u,v\) would vanish there. Unique continuation for smooth Laplace eigenfunctions on the connected sphere forces it to vanish everywhere [1], contradicting orthonormality. Independence of \(U_1,U_2\) makes their Gram matrix for \(d\mu_w\) positive definite. Testing in the directions \(U_1,U_2\), for example, proves rank two. When \(u,v\) have the same parity, both functions in (8) are even, so the same tests are available in the even family.

Finally, conjugating the Laplacian by the parameter-dependent pullback adds to its variation a commutator with the Laplacian. The compression of such a commutator to an eigenspace on which the Laplacian is \(\lambda I\) is zero. Thus the coordinate change does not change the splitting matrix. The zero eigenvalue requires no splitting argument: it is simple because \(S^2\) is connected. ◻

Proof. We give the finite-dimensional reduction, including the treatment of clusters of multiplicity greater than two. The use of spectral projections and local cluster matrices is the usual perturbation construction [11]. In the fixed-area gauge the operators have common domain \(H^2(S^2)\) in \(L^2(\mu)\) and depend smoothly, as maps \(H^2\to L^2\), on the finite parameters. On a contour in the resolvent set, elliptic estimates give bounded inverses \(L^2\to H^2\); the inverse-difference identity and a Neumann series give smooth parameter dependence. Integrating the resolvents around the contour therefore gives a smooth finite-rank projection. This argument needs smooth parameter dependence, rather than analytic eigenvalue branches. The avoidance step is a finite-dimensional general-position argument of the type used in spectral genericity [16].

Start with a smooth preliminary path \(w_0(t)\), \(0\leq t\leq1\), joining the two parameters and constant near them. Convexity of \(\mathcal U\) supplies such a path. The endpoints have positive gaps through the cutoff, so a small initial and final time interval remains simple under all sufficiently small perturbations of the path. Choose a smooth cutoff \(\chi(t)\) that vanishes near \(0,1\) and equals \(1\) outside these two simple intervals.

Fix the cutoff \(q\). Uniform comparison with the round mass form bounds \(\lambda_{q+1}(h^0(w))\) above throughout a small neighborhood of the path. Choose a larger number \(\Lambda\). Every repeated cluster that can affect simplicity through \(q\) has eigenvalue below \(\Lambda\), including all members of a cluster that straddles the cutoff. The reverse form comparison bounds the number of eigenvalues below \(\Lambda\) uniformly. Consider all triples \[(t,u,v),\qquad -\Delta_{h^0(w_0(t))}u=\lambda u,\quad -\Delta_{h^0(w_0(t))}v=\lambda v,\quad 0<\lambda\leq\Lambda,\] where \(u,v\) are real and orthonormal for \(d\mu_{w_0(t)}\). This set of normalized pairs is compact in the smooth topology. To see the needed compactness directly, write the eigenvalue equation as \(-\Delta_0u=\lambda\rho_{w_0(t)}u\). The coefficients have uniform smooth bounds along the compact path, the mass norms are uniformly equivalent, and \(\lambda\) is bounded. Elliptic estimates followed by Sobolev embedding give bounds for every spatial derivative. A diagonal subsequence then converges smoothly, and the equation and orthonormality pass to the limit. The first positive eigenvalue is uniformly bounded away from zero, so the limit is again a positive eigenvalue pair. In the even construction take only pairs of the same specified parity; this is a closed condition. If there is no such pair, the preliminary path is already simple through the cutoff and no perturbation is needed.

At each such pair, Lemma 3 supplies two smooth directions for which the two traceless entries have an invertible derivative. This property is open in \((t,u,v)\). A finite cover of the compact set of pairs therefore supplies a single finite list of directions \(\xi_1,\ldots,\xi_D\) whose span \(E\) has rank two on every one of these pairs. In the parity version all \(\xi_j\) are even. The same rank assertion holds for relevant eigenfunction pairs at all parameters in a sufficiently small \(E\)-neighborhood of the path. Otherwise there would be a sequence of counterexamples with parameters tending to the path. The same elliptic compactness would give a limiting pair on the path, where one of the selected two-by-two determinants is nonzero, a contradiction.

Use this fixed pool in the finite family \[ w(t,z)=w_0(t)+\chi(t)\sum_{j=1}^D z_j\xi_j, \qquad z=(z_1,\ldots,z_D). \tag{9}\] For small \(z\) it stays in \(\mathcal U\); the endpoint intervals remain simple. At every possible bad point outside those intervals, \(\chi=1\), and the two traceless entries have rank two in the \(z\) variables.

We now describe precisely the sets to be avoided. At a bad point \((t_*,z_*)\), let \(r,\ldots,s\) be the entire consecutive block occupied by one repeated eigenvalue, with strict spectral gaps immediately outside that block. This includes \(s>q+1\) if the cutoff meets a larger cluster. A contour enclosing this eigenvalue and no other spectrum gives, in a neighborhood of \((t_*,z_*)\), a smooth Riesz projection of rank \(m=s-r+1\). Projecting a basis at the center and orthonormalizing gives a smooth real frame; elliptic regularity makes it smooth in the spatial variables as well. In this frame the operator on the cluster is a smooth symmetric matrix \(M(t,z)\). Choose the first two frame vectors at the center to be any orthonormal pair in the repeated eigenspace and set \[ F(t,z)=\bigl(M_{11}(t,z)-M_{22}(t,z),\,2M_{12}(t,z)\bigr). \tag{10}\] At the center the derivative of \(F\) in \(z\) has rank two, by the choice of \(E\) and (7). Shrinking the neighborhood keeps this rank. The implicit-function theorem makes \(F^{-1}(0)\) there a smooth codimension-two submanifold of time times parameter space.

If at a point in this neighborhood the same block \(r,\ldots,s\) is still entirely equal, then \(M\) is scalar and \(F=0\). This is the only containment asserted for this chart. When an \(m\)-fold block partly splits but leaves a smaller repeated block, the smaller block has its own chart centered at that point. Thus the two equations in (10) handle every \(m\geq2\): they contain the locus of an exact \(m\)-fold block without purporting to describe all partial repetitions of the larger Riesz cluster.

For each fixed pair of indices \(r,s\), take the stratum on which \(r,\ldots,s\) is the entire repeated block, with strict gaps outside it. This stratum, as a subspace of the second-countable space \([0,1]\times\mathbb R^D\), has a countable subcover by neighborhoods centered on that same stratum. Every point of the stratum in one of these neighborhoods lies in that chart’s zero set, because the same block is entirely equal there. Taking the union over the countably many possible pairs \(r,s\) covers all bad points by countably many codimension-two zero sets. Each zero set projects to a null set in \(\mathbb R^D\). For example, in a local implicit-function chart two coordinates of \(z\) are smooth functions of \(t\) and the remaining \(D-2\) coordinates. Its projection is locally a smooth, hence locally Lipschitz, image of a space of dimension \(D-1\) in \(\mathbb R^D\), and has \(D\)-dimensional Lebesgue measure zero. A countable union still has measure zero. There are therefore arbitrarily small choices of \(z\) whose path meets none of the bad sets. Formula (9) fixes the endpoints.

The endpoint density assertion is the same argument without the time variable. Begin at one parameter, choose the finite pool for its normalized low-eigenvalue pairs, and keep its rank in a small neighborhood. Each exact-block zero set then has codimension two in the finite parameter space itself, so the union of the bad sets is null and its complement is dense. In the even family, perform all of these constructions separately for the two parity lists with the common finite pool of even directions. The union of their bad sets is still countable and null. This proves both versions. ◻

Proof. Let \(u,v\) be an orthonormal basis for the chosen double at the current parameter, and put \(U_1=u^2-v^2\), \(U_2=2uv\). The linearized constraints for keeping its cluster matrix scalar have kernel \[ \mathcal K=\left\{\xi\in C^\infty(S^2;\mathbb R): \int U_1\xi\,d\mu_w=\int U_2\xi\,d\mu_w=0\right\}. \tag{11}\] This has codimension two in the full space of smooth conformal directions, by Lemma 3. We first prove that, for any different repeated positive eigenspace, some direction in \(\mathcal K\) has a nonscalar compression to that eigenspace.

Suppose instead that every \(\xi\in\mathcal K\) had scalar compression on such an eigenspace, of eigenvalue \(\mu_*>0\). Choose an orthonormal pair \(a,b\) in it. By (7), the two independent functionals represented by \[A_1=a^2-b^2,\qquad A_2=2ab\] would both vanish on \(\mathcal K\). A linear functional that vanishes on the kernel of the surjection \(\xi\mapsto(\int U_1\xi\,d\mu_w,\int U_2\xi\,d\mu_w)\) is a linear combination of those two component functionals. Since the two \(A\)-functionals are themselves independent, their span must equal the span of the two \(U\)-functionals. Equality of the functionals on every smooth \(\xi\), and positivity of \(\rho_w\), then gives the pointwise identity \[ Q(a,b)=TQ(u,v),\qquad Q(x,y)=(x^2-y^2,\,2xy), \tag{12}\] for a constant invertible real \(2\times2\) matrix \(T\). There is no undetermined additive constant here: each of the four product functions has mean zero. This argument took place in the full smooth tangent space before any finite-dimensional restriction.

Identity (12) forces an equality of nodal sets. In fact \(a=0\) is equivalent to \(Q(a,b)\in\{(-t,0):t\geq0\}\). The inverse image of this closed ray under \(T\) is another closed ray through the origin. Identifying \(\mathbb R^2\) with \(\mathbb C\), the map \(Q\) is the squaring map, so the inverse image under \(Q\) of a closed ray is one unoriented line. There is consequently a nonzero pair \((c,d)\) such that \[ \{a=0\}=\{cu+dv=0\}. \tag{13}\] The assertion includes points where both entries of either pair vanish, because the rays include the origin.

Here is why (13) is impossible for distinct eigenvalues. Write \(f=cu+dv\), a nonzero eigenfunction for the chosen eigenvalue \(\lambda_*\), and take a connected component \(\Omega\) of the common nonzero set. Change the signs so that both \(a\) and \(f\) are positive on \(\Omega\). Each restriction belongs to \(H^1_0(\Omega)\), without any regularity assumption on the nodal boundary. For example, \[f_\varepsilon=(f-\varepsilon)_+\] has compact support in \(\Omega\): its support stays away from the boundary, where \(f=0\). As \(\varepsilon\downarrow0\), these truncations converge to \(f\) in \(H^1(\Omega)\), by dominated convergence for the functions and their gradients. Each truncation can be approximated in \(H^1\) by smooth functions compactly supported in \(\Omega\). The same reasoning applies to \(a\). Their weak eigenvalue equations may therefore be tested against one another on \(\Omega\). Symmetry of the Dirichlet form gives \[(\mu_*-\lambda_*)\int_\Omega af\,d\mu_w=0.\] The integral is positive. The two eigenvalues would be equal, contrary to the choice of a different spectral cluster. This proves the existence of a nonscalar direction \(\xi\in\mathcal K\).

We now realize that tangent direction while keeping the chosen double exactly. Pick two smooth directions \(\zeta_1,\zeta_2\) on which the derivative of its two traceless entries is invertible. In the three-parameter family \[w(t,x_1,x_2)=w+t\xi+x_1\zeta_1+x_2\zeta_2\] form the smooth \(2\times2\) Riesz cluster matrix \(M_{\rm d}\) of the chosen double and its two constraints \(F_{\rm d}=(M_{{\rm d},11}-M_{{\rm d},22},2M_{{\rm d},12})\). They vanish exactly when that cluster is scalar. The derivative in \((x_1,x_2)\) is invertible, so the finite-dimensional implicit-function theorem gives smooth \(x_1(t),x_2(t)\) with \(F_{\rm d}=0\). Because \(\xi\in\mathcal K\), their derivatives at \(0\) are zero. The resulting curve has tangent \(\xi\). On the other repeated cluster its matrix is therefore \[\mu_* I+tD+o(t)\] with \(D\) nonscalar. For a sufficiently small nonzero \(t\), the distinct eigenvalues of \(D\) separate that cluster into at least two smaller groups. The chosen cluster remains a double.

Only finitely many repetitions must be removed. Initially extend the list past \(q+1\) to the last member \(q'\) of the cluster containing \(\lambda_{q+1}\). There is a strict gap after \(q'\). At each step take the curve parameter small enough to preserve that gap, the isolation of the chosen double, and every strict gap already obtained among these \(q'\) positions. If the multiplicities of the current clusters are \(m_1,\ldots,m_r\), the integer \(\sum_j\binom{m_j}{2}\) strictly decreases whenever an unwanted cluster is split, while the contribution of the chosen double stays equal to \(1\). Thus finitely many steps separate every other cluster. The steps may be chosen with arbitrarily small total size, and the isolation gaps keep the chosen double at positions \(i,i+1\). The claimed neighborhood and cutoff conclusions follow. ◻

Proof. The separated equation for a profile \(v\) is \[-(\sin\theta\,v')'+\frac{m^2}{\sin\theta}v =\lambda\zeta^2\sin\theta\,v.\] Its form domain is the closure of smooth sphere sector profiles in the energy-plus-mass norm. Compact spectral theory on the smooth compact sphere, restricted to this closed sector, supplies its eigenfunctions and the variational minimum. Their profiles are smooth at the poles: for \(m\geq1\) they have the form \(t^m a(t^2)\) in polar distance \(t\), and for \(m=0\) they are smooth even functions of \(t\). For two profiles \(u,v\) at the same eigenvalue, the equation makes \(W=\sin\theta(uv'-u'v)\) constant. The stated pole behavior makes \(W\to0\) at a pole: for \(m\geq1\), \(u,v=O(t^m)\) and their derivatives are \(O(t^{m-1})\); for \(m=0\), the functions are bounded and their derivatives are \(O(t)\). Thus \(W=0\). At any interior point where \(u\) is nonzero, \(v\) and a scalar multiple of \(u\) have equal value and derivative. Interior ODE uniqueness makes them equal everywhere.

Replacing a minimizing real profile by its absolute value preserves the form domain and its energy and mass. The resulting nonnegative minimizer satisfies the weak equation and is smooth in the open interval. An interior zero would be a minimum with zero derivative, so ODE uniqueness would force the profile to vanish. The ground profile is consequently positive. Reflection preserves the equation and pole conditions. Simplicity and positivity force the mass-normalized ground profile to be reflection even. With its azimuthal factor its antipodal parity is \((-1)^m\).

For \(m=0\), simplicity holds at every position by the same Wronskian argument. Min–max gives continuous ordered eigenvalues along a smooth deformation, and a local simple spectral projection gives a continuous unit eigenfunction. Reflection acts on its line by \(+1\) or \(-1\); this sign is continuous and hence constant. At round it is the sign of the corresponding Legendre polynomial, \((-1)^l\). No nodal count for a singular interval problem is required. ◻

Proof. Pairs inside one round block. Suppose that \(i,i+1\) belong to the round degree-\(l\) block. Multiplication of harmonic functions defines \[ \operatorname{Sym}^2\mathcal Y_l \longrightarrow\bigoplus_{j=0}^l\mathcal Y_{2j}, \qquad u\mathbin{\odot}v\longmapsto uv. \tag{14}\] The target contains the products because a homogeneous polynomial of degree \(2l\) decomposes, on the sphere, into harmonic components of degrees \(2l,2l-2,\ldots,0\). The map is an isomorphism. This is the round-sphere product-space fact in Greilhuber–Kepplinger [9]; the following argument records the particular algebra that we need.

Its image is invariant under rotations, hence under the rotation Casimir \(\Delta_0\). It contains \[F_l(z)=(1-z^2)^l =\bigl(\operatorname{Re}((x+\mathrm i y)^l)\bigr)^2 +\bigl(\operatorname{Im}((x+\mathrm i y)^l)\bigr)^2 \quad\text{on }S^2 .\] For functions of \(z\), \(\Delta_0=(1-z^2)\partial_z^2-2z\partial_z\), and direct differentiation gives \[ \bigl(\Delta_0+2l(2l+1)\bigr)F_l=4l^2F_{l-1}. \tag{15}\] Expand \(F_l=\sum_{j=0}^l c_{lj}P_{2j}(z)\) in Legendre polynomials. Its top coefficient is nonzero by polynomial degree, and for \(j<l\), (15) gives \[c_{lj}= \frac{4l^2}{2l(2l+1)-2j(2j+1)}\,c_{l-1,j}.\] Starting with \(F_0=1\), every \(c_{lj}\) is therefore nonzero. Polynomial spectral projections in \(\Delta_0\) put a nonzero zonal vector of each \(\mathcal Y_{2j}\) in the image of (14). Rotations of a nonzero zonal harmonic span its harmonic space: they are the harmonic projections of powers of linear forms, and such powers span the homogeneous polynomials. The map is consequently onto. Its source and target both have dimension \[\frac{(2l+1)(2l+2)}2 =\sum_{j=0}^l(4j+1)=(l+1)(2l+1),\] so it is an isomorphism.

Dualizing this isomorphism in the variation formula (7) shows that the trace-free first-variation map for the entire round degree-\(l\) cluster is onto the trace-free symmetric matrices on \(\mathcal Y_l\). More explicitly, a nonzero trace-free matrix \(C\) gives the nonzero product function \(\sum C_{\alpha\beta}u_\alpha u_\beta\) by injectivity of (14), so it cannot annihilate every conformal direction. Choose finitely many directions on which this derivative is onto, and represent the cluster by its smooth Riesz matrix in that finite family. The submersion theorem realizes every sufficiently small trace-free matrix as its trace-free part. Choose a diagonal one whose ordered entries are strictly increasing except for the desired adjacent equality. It may be made arbitrarily small. The round gaps to the other degree blocks remain positive, so this is an isolated double at the required positions. Lemma 5, with cutoff \(p+1\), separates all other repetitions while retaining it.

Pairs across consecutive round blocks. These pairs have \(i=B_l\) with \(L+1\leq l\leq M-1\). Put \(P=(-1)^l\) and \(c_0=3/2\). Choose once and for all a smooth axisymmetric function \(\psi(\theta)\), supported in an equatorial band whose closure is disjoint from the poles, where \(\theta\) is colatitude, such that \[ \psi(\pi-\theta)=\psi(\theta),\qquad 0\leq\psi\leq1,\qquad \psi(\pi/2)=1,\qquad \bar\psi=\frac1\pi\int_0^\pi\psi(\theta)\,d\theta<\frac1{10}. \tag{16}\] For large \(l\), consider the unnormalized metric \[ \widehat h_l=\zeta_l(\theta)^2g_0,\qquad \zeta_l=1-\frac{c_0}{l}\psi. \tag{17}\] It is antipodally invariant and tends to \(g_0\) in every fixed smooth norm. It corresponds to the conformal parameter \(w_l=\log\zeta_l\). Dividing the metric by its area factor multiplies all its eigenvalues by the same positive number, so it does not change any ordering used below.

The Dirichlet form is unchanged and \((1-c_0/l)^2\leq\zeta_l^2\leq1\). Min–max on either parity subspace therefore places the positions belonging at round to degree \(q\) in the interval \[ \bigl[q(q+1),\,(1-c_0/l)^{-2}q(q+1)\bigr]. \tag{18}\] The intervals for \(q\) and \(q+2\) in the same parity are disjoint through \(q=l+1\), for all large \(l\). Indeed the gap between their round values is \(4q+6\), while, uniformly for \(q\leq l+1\), \[\bigl((1-c_0/l)^{-2}-1\bigr)q(q+1) \leq (2c_0+o(1))(q+1)<4q+6.\] Here \(c_0<2\) gives the last inequality uniformly for large \(l\). Thus the indicated degree groups keep their order inside each parity list.

We next find a value from the parity-\(P\) degree-\(l\) group above a value from the parity-\(-P\) degree-\((l+1)\) group. The sector min–max statements use the Friedrichs form domains obtained by closing smooth sphere sector profiles in their energy-plus-mass norms. Near either pole, a smooth sphere sector profile of azimuthal number \(m\) has the form \(t^{|m|}a(t^2)\), where \(t\) is distance to that pole and \(a\) is smooth; for \(m=0\), these are smooth profiles even at the poles. For the first value, take the lowest eigenvalue in azimuthal sector \(m=l\) of \(\widehat h_l\). By Lemma 6, the meridional ground profile is positive and unique up to scale. Reflection in the equator preserves it; the azimuthal factor acquires \((-1)^l\) under the antipodal map. Hence the resulting mode has parity \(P\). Choose a real eigenfunction \(u_l\) with \(\int u_l^2\,d\mu=1\), and write its round energy and changed mass as \[E_l=\int_{S^2}|\nabla_0u_l|^2\,d\mu,\qquad M_l=\int_{S^2}\zeta_l^2u_l^2\,d\mu.\] The azimuthal term gives \[E_l\geq l^2\int_{S^2}\frac{u_l^2}{\sin^2\theta}\,d\mu, \qquad M_l\leq1.\] Min–max in this sector bounds \[\frac{E_l}{M_l}\leq(1-c_0/l)^{-2}l(l+1)=l^2+O(l).\] It follows that \(\int(\sin^{-2}\theta-1)u_l^2\,d\mu=O(l^{-1})\). The integrand factor is bounded below by a positive constant away from every fixed equatorial neighborhood. The round probability masses \(u_l^2d\mu\) therefore concentrate at the equator. Since \(\psi=1\) there and is continuous, \[\int\psi u_l^2\,d\mu=1-o(1),\qquad M_l=1-\frac{2c_0}{l}+o(l^{-1}).\] The lowest round eigenvalue in the sector is \(l(l+1)\), so also \(E_l\geq l(l+1)\). The chosen high value satisfies \[ \lambda_{\rm high}=\frac{E_l}{M_l} \geq l(l+1)\bigl(1+2c_0/l+o(l^{-1})\bigr) =l^2+(1+2c_0)l+o(l). \tag{19}\] Its bounds place it in the degree-\(l\) group of the parity-\(P\) list.

For the low value use the axisymmetric eigenfunction in the slot occupied at round by degree \(l+1\), namely slot \(l+2\) when this list is numbered from \(1\). Lemma 6 shows that its eigenvalue is simple throughout the reflection-symmetric deformation and that its reflection parity remains \((-1)^{l+1}=-P\).

Set \[R=\frac1\pi\int_0^\pi\zeta_l(\theta)\,d\theta =1-\frac{c_0\bar\psi}{l},\qquad x(\theta)=\frac1R\int_0^\theta\zeta_l(t)\,dt .\] Then \(x\) maps \([0,\pi]\) onto itself, and in this coordinate \[\widehat h_l=R^2\bigl(dx^2+\sin^2x\,F(x)^2d\varphi^2\bigr), \qquad F(x)=\frac{\zeta_l(\theta(x))\sin\theta(x)}{R\sin x}.\] The function \(F\) is positive and reflection-symmetric. It has smooth even extensions at \(0,\pi\) and satisfies \(F=1+O(l^{-1})\) in every fixed smooth norm. For example, \(\zeta_l=1\) near a pole, so there \(\theta=Rx\) and \(F=\sin(Rx)/(R\sin x)\), which has the stated even extension and bounds. The other pole follows by reflection.

The axisymmetric nonnegative Laplacian of the metric in parentheses is \[-\frac1{\sin x\,F}\partial_x(\sin x\,F\,\partial_x)\] on \(L^2(\sin x\,F\,dx)\). We use the Friedrichs realizations of the Dirichlet forms defined first on smooth profiles with even smooth extensions at both poles. Multiplication by \(\sqrt F\) maps this core bijectively to the corresponding round core, since \(\sqrt F\) and its reciprocal are smooth and even at the poles. On this core, the boundary contribution \(\bigl[\sin x\,(\sqrt F)'(\sqrt F)^{-1}|v|^2\bigr]_0^\pi\) vanishes. The conjugated form is therefore the round axisymmetric form on \(L^2(\sin x\,dx)\) plus the potential \[V_F=\frac{(\sqrt F)''+\cot x\,(\sqrt F)'}{\sqrt F}.\] The even endpoint extensions make the apparent endpoint singularities removable, and \(\|V_F\|_\infty=O(l^{-1})\). This bounded potential makes the shifted form norms equivalent, so multiplication by \(\sqrt F\) also maps the closed form domain onto the round form domain. Min–max for a bounded potential, at the degree-\((l+1)\) slot, gives \[ \begin{aligned} \lambda_{\rm low} &=R^{-2}\bigl((l+1)(l+2)+O(l^{-1})\bigr)\\ &=l^2+(3+2c_0\bar\psi)l+O(1). \end{aligned} \tag{20}\] The same form comparison on the axisymmetric subspace places this slot in the interval (18) for \(q=l+1\). The separated parity intervals therefore locate it in the degree-\((l+1)\) group of the parity-\(-P\) list. Combining (19) and (20) yields the strict inversion \[ \lambda_{\rm high}-\lambda_{\rm low} \geq\bigl(2c_0(1-\bar\psi)-2\bigr)l+o(l) =(1-3\bar\psi)l+o(l)>0. \tag{21}\]

To locate the resulting double in the combined list, write \(\lambda_j^P\) and \(\lambda_j^{-P}\) for the ordered parity eigenvalues. The counts up to round degree \(l\) are \[ A_l=\frac{(l+1)(l+2)}2\quad\text{in parity }P,\qquad D_l=\frac{l(l+1)}2\quad\text{in parity }-P,\qquad A_l+D_l=B_l. \tag{22}\] At the round metric, \[\lambda_{A_l}^P=l(l+1) <(l+1)(l+2)=\lambda_{D_l+1}^{-P}.\] At \(\widehat h_l\), the high value lies among the first \(A_l\) values of parity \(P\), while the low value lies in the group starting at \(D_l+1\) in parity \(-P\). Hence \[\lambda_{A_l}^P\geq\lambda_{\rm high} >\lambda_{\rm low}\geq\lambda_{D_l+1}^{-P}.\] These strict indexed inequalities survive area normalization and sufficiently small perturbations of the two endpoints. By Lemma 4, choose such endpoints that are simple through the needed cutoffs within each parity, and join them by an even path with that property. Along it the continuous function \(\lambda_{A_l}^P-\lambda_{D_l+1}^{-P}\) changes sign. At a zero, the two equal values are simple in their separate parity lists. Exactly \((A_l-1)+D_l=B_l-1\) eigenvalues of the combined list are below them. Thus the equality is a double at positions \(B_l,B_l+1\). This conclusion requires only continuity of the indexed parity eigenvalues; no isolation of coincidences along this preliminary path is needed. Apply Lemma 5 in unrestricted directions to keep this double and separate every other level through \(p+1\).

The two cases cover all adjacent pairs in \(I\). For fixed \(k\) there are finitely many of them. All cross-block estimates above are estimates as \(l\to\infty\), so they hold simultaneously for \(L+1\leq l\leq M-1\) once \(L\) is large. The cross-block parameters \(w_l=\log(1-c_0\psi/l)\) tend smoothly to zero uniformly for \(l\geq L+1\) as \(k\) tends to infinity; the remaining perturbations can be arbitrarily small. Thus one prescribed \(\mathcal U\) contains all the metrics selected for every sufficiently large \(k\). ◻

Proof. For smooth data, the classical Dirichlet theorem gives a smooth harmonic extension \(u\) inside the ball [8]. The fixed angular area form makes its equation away from the origin \[u_{rr}+2\frac{f'}f u_r+f^{-2}\Delta_{H_z(\log r)}u=0.\] Consequently, for \(\bar u(r)=\int u(r,\omega)\,d\mu(\omega)\), \[(f(r)^2\bar u_r(r))'=0.\] Regularity in the Euclidean core and \(f(r)=r\) near zero force the constant of integration to vanish. The mean on every inner sphere therefore equals the boundary mean. The maximum principle makes harmonic restriction positive and constant preserving [8]. A positive constant-preserving map satisfies the pointwise square inequality \((\mathcal R v)^2\leq\mathcal R(v^2)\): apply positivity to \(\mathcal R((v-c)^2)\) with \(c=\mathcal Rv\) at the point in question. After integration and mean preservation this gives \[ \|\mathcal R^z_{r,R}v\|_2^2 \leq\int\mathcal R^z_{r,R}(v^2)\,d\mu =\int v^2\,d\mu. \tag{46}\] Density extends the maps to \(L^2(\mu)\). For clarity, these extended maps still are traces of one harmonic function in the interior: approximate the boundary value in \(L^2\) by smooth data and use (46) on inner spheres. The local \(L^2\)-to-sup estimate, followed by interior Schauder estimates and their iterates for the differentiated smooth equation, makes the extensions Cauchy in every smooth norm on smaller compact sets [8]. Their limit is harmonic, and uniqueness follows from the same approximation. This also shows that restriction from \(L^2\) boundary data is smooth on every strictly inner sphere.

In logarithmic radius \(t=\log r\), the equation becomes \[ u_{tt}+\gamma(t)u_t+b(t)^{-2}\Delta_{H_z(t)}u=0, \qquad \gamma(t)=1+2b_t(t)/b(t). \tag{47}\] For the local estimates just used and those below, its divergence form is \[\partial_t(e^t b(t)^2u_t)+e^t\Delta_{H_z(t)}u=0.\] In fixed angular charts the second term is divergence with respect to the fixed density \(\mu\). On a time interval of fixed width, division by \(e^{t_0}\) at its center leaves uniformly elliptic coefficients with the uniform smooth bounds in the input. The cited local estimates therefore have constants independent of the large center time and of the controls. The compact core is covered by ordinary smooth coordinate charts. We first prove the operator-norm statement. Consider arbitrary \(j_n\to\infty\), arbitrary allowed starts \(J_n\leq j_n\) and control sequences, and data \(\|v_n\|_2\leq1\). Pass to a subsequence for which \(v_n\rightharpoonup v\) weakly and \(H(s(j_n))\to H_\infty\) smoothly. Such a phase subsequence exists because the reference program is compact. Move the outer boundary to zero by \(x=t-(j_n+1)\), and write \(u_n(x)\) for the exact whole-ball extension. On every fixed compact interval in \(x\leq0\), the radial and angular input estimates make the shifted coefficients converge smoothly to those of \[ U_{xx}+U_x+a^{-2}\Delta_{H_\infty}U=0. \tag{48}\] Indeed, the phase changes by \(O(\eta(j_n))\) on such an interval and the control amplitudes vanish. The interval on which the stated end bounds hold extends an unbounded distance to the left in these coordinates, since \(j_n-0.8J_n\geq0.2j_n\) when \(J_n\leq j_n\).

The sectional bounds \(\|u_n(x)\|_2\leq1\) hold for every \(x<0\). On each compact subcylinder of \(x<0\) they give an \(L^2\) bound for the solution. The local \(L^2\)-to-sup estimate and interior Schauder estimates [8] give uniform \(C^{2,\alpha}\) bounds on smaller cylinders. Compact embedding, followed by a diagonal subsequence, gives strong \(L^2\) convergence at each interior sphere, in particular at \(x=-1\). The limit \(U\) solves (48) on \((-\infty,0)\) and has sectional norm at most one there.

The trace of this limit at \(x=0\) must also be identified. For a fixed smooth \(\phi\) on \(S^2\), put \(y_n(x)=\langle u_n(x),\phi\rangle_2\). Self-adjointness of the angular Laplacian in the fixed space \(L^2(\mu)\) gives, on \([-1,0)\), \[y_n''+\gamma_n y_n' =-\langle u_n,b_n^{-2}\Delta_{H_{z,n}}\phi\rangle_2.\] The right side and \(y_n\) are uniformly bounded, and so is \(\gamma_n\). The mean value theorem on \([-1,-1/2]\) bounds \(y_n'\) at one point. Integrating the first-order equation for \(y_n'\) with its bounded integrating factor then bounds \(y_n'\) on all of \([-1,0)\). For smooth boundary data \(y_n(0)=\langle v_n,\phi\rangle_2\); the same identity and Lipschitz bound pass to \(L^2\) data by the preceding approximation. Taking the limit first in \(n\) and then in \(x\uparrow0\) shows that \(U\) has weak boundary trace \(v\).

For an eigenvalue \(\lambda\geq0\) of \(-\Delta_{H_\infty}\), the two radial roots in (48) are \(\mathfrak d(a^{-2}\lambda)\) and \(-1-\mathfrak d(a^{-2}\lambda)\). The coefficient of the latter root must vanish because \(U\) remains sectionally \(L^2\)-bounded as \(x\to-\infty\). This also excludes the root \(-1\) on constants. The weak trace fixes the remaining coefficient of every eigenfunction, so \[U(x)=\exp[x\mathfrak d(a^{-2}(-\Delta_{H_\infty}))]v,\qquad x\leq0.\] Thus \(T_{j_n}v_n\) converges strongly to \(U(-1)\). Apply the same argument to the frozen half-cylinder solutions \[U_n^0(x)=\exp[x\mathfrak d(b(j_n)^{-2}L(s(j_n)))]v_n.\] They have the same sectional bound, the same weak boundary trace limit, and the same frozen limiting equation; hence \(P_{j_n}^0v_n=U_n^0(-1)\) has the same strong limit. If (44) failed, one could choose such \(v_n\) nearly attaining a fixed positive lower bound for \(\|T_{j_n}-P_{j_n}^0\|\), contradicting these two strong limits. The sequence of controls and starts was arbitrary, which proves the stated uniformity.

For the finite-frame estimate write \(\eta_j=\eta(j)\) and set \[\ell_j=4\log(1/\eta_j)=3\log j,\qquad \varepsilon_j=(1+\ell_j)\eta_j.\] Take \(J\) late enough that \(j-\ell_j>0.8J\) for every \(j\geq J\). It suffices to check this at \(j=J\), since \(j-3\log j\) is increasing for \(j>3\). The entire comparison cylinder \([j-\ell_j,j+1]\) then lies in the controlled end, including the first step. For \(\phi_j=e_i(s(j+1))\), let \[U_j(t)=\exp[(t-j-1)\mathfrak d(b(j)^{-2}(-\Delta_{H_z(j)}))]\phi_j, \qquad t\leq j+1.\] This is the bounded-backward solution of the frozen equation with boundary value \(\phi_j\). The vectors \(\phi_j\) have uniform smooth norms, and the frozen metrics range over a compact smooth family. Applying powers of their Laplacians in the spectral expansion bounds all required spatial and time Sobolev norms of \(U_j\), uniformly for \(t\leq j+1\). Elliptic estimates on the sphere and Sobolev embedding therefore give the required uniform pointwise derivative bounds, including \(\|U_j(t)\|_\infty\leq C\).

On \([j-\ell_j,j+1]\), speeds are comparable to \(\eta_j\). By (41)–(42), the angular coefficients vary from their values at \(j\) by \(O((1+\ell_j)\eta_j)\) in every needed spatial norm, and \(\gamma(t)-1=O(\eta_j^2)\). Substituting \(U_j\) into the actual operator in (47) thus leaves a residual \(R_j\) with \(\|R_j\|_\infty\leq C\varepsilon_j\). Let \(u_j\) be the exact whole-ball extension of \(\phi_j\), and put \(w_j=u_j-U_j\) on this cylinder. At \(j+1\) it vanishes. At \(j-\ell_j\) it is uniformly bounded in sup norm: the maximum principle bounds \(u_j\) by \(\|\phi_j\|_\infty\) throughout its ball, independently of every earlier control, and \(U_j\) has the bound just obtained.

For late \(J\), \(\gamma\geq3/4\). Choose one sufficiently large constant \(K\), and define \[B_j(t)=K\varepsilon_j(j+1-t) +K\exp[-(t-j+\ell_j)/2].\] It dominates \(|w_j|\) at both ends. Since \[(\partial_t^2+\gamma\partial_t)B_j =-K\gamma\varepsilon_j+ K(1/4-\gamma/2)\exp[-(t-j+\ell_j)/2] \leq-C\varepsilon_j,\] the comparison principle applied to \(B_j+w_j\) and \(B_j-w_j\) [8] gives \[\|w_j(j)\|_2\leq\|w_j(j)\|_\infty \leq C(\varepsilon_j+e^{-\ell_j/2}) =O\left(\eta_j\log(1/\eta_j)+\eta_j^2\right).\] The last expression divided by \(\delta_j=\eta_j^{3/4}\) tends to zero. This proves (45). Only the local coefficient bounds entered the residual; the whole-ball maximum principle absorbed the entire earlier history into the bounded left discrepancy. ◻

Proof. The frame moves by \(O(\eta(j))=o(\delta_j)\) over one step. The perturbed first-\(p\) spectral projection \(\Pi_j^*\) for \(H_z(j)\) satisfies \(\|\Pi_j^*-\Pi_j\|=O(\delta_j)\): apply the isolated-cluster projection formula to (42) and (40). This assertion concerns the entire cluster, so internal double crossings cause no loss. Since \(P_j^*\) preserves \(\Pi_j^*\), its action on the reference frame has high component \(O(\delta_j)\). Smooth finite-cluster functional calculus gives its low block as \(\operatorname{diag}(\alpha_i(j)^{-1})+O(\delta_j)\). The finite-frame estimate (45), together with frame movement, proves the first two bounds in (49).

The other two blocks need only (44). The reference operator \(P_j^0\) preserves \(E_j\) and \(Q_j\); replacing \(Q_{j+1}\) by \(Q_j\) costs \(O(\eta(j))\). Hence \(T_{LQ}=o(1)\). The gap (40) and strict monotonicity of \(\mathfrak d\) give a fixed \(\kappa_{\mathrm{ref}}<1\) such that \[\|P_j^0|_{Q_j}\| \leq \kappa_{\mathrm{ref}}\tau_j .\] Indeed the difference between \(\mathfrak d(b^{-2}\lambda_{p+1}^{\mathrm{ord}})\) and the largest low frequency has a positive minimum over the compact phase and \(b\in[a,1]\). Choose \(\kappa_{\mathrm{ref}}<\kappa<1\). The absolute \(o(1)\) error in (44) and the frame movement can be absorbed into \((\kappa-\kappa_{\mathrm{ref}})\tau_j\) because \(\tau_j\geq\tau_*>0\). This is where fixing \(p\) before \(J\) is essential.

For \(V:\mathbb R^p\to Q_{j+1}\) in a fixed small operator ball, define \[M_j(V)=T_{LL}+T_{LQ}V,\qquad \mathcal F_j(V)=(T_{QL}+T_{QQ}V)M_j(V)^{-1}.\] If \(D_j=\operatorname{diag}(\alpha_i(j)^{-1})\), the relative perturbation \[\|D_j^{-1}(M_j(V)-D_j)\| \leq\tau_*^{-1}\bigl(O(\delta_j)+o(1)\|V\|\bigr)\] tends uniformly to zero. Thus \(M_j(V)\) is invertible and \(\|M_j(V)^{-1}\|\leq\tau_j^{-1}(1+o(1))\) throughout that ball. In particular \(\|\mathcal F_j(0)\|\leq C\delta_j\). The inverse-difference identity gives \[\begin{align*} \mathcal F_j(V)-\mathcal F_j(W) ={}&T_{QQ}(V-W)M_j(V)^{-1}\\ &+(T_{QL}+T_{QQ}W) \bigl(M_j(V)^{-1}-M_j(W)^{-1}\bigr),\\ M_j(V)^{-1}-M_j(W)^{-1} ={}&-M_j(V)^{-1}T_{LQ}(V-W)M_j(W)^{-1}. \end{align*}\] The first term has Lipschitz constant at most \(\kappa+o(1)\). The second has constant \(o(1)\), since the ball is fixed, \(\tau_*>0\), and \(T_{LQ}=o(1)\). Choose \(\kappa<\kappa_0<1\), and then take \(J\) late enough that every \(\mathcal F_j\) has Lipschitz constant at most \(\kappa_0\). Increasing \(J\) once more makes \(\|\mathcal F_j(0)\|\) smaller than \((1-\kappa_0)\) times the ball radius, so every transform maps that ball into itself.

For an outer cut \(N\), start with \(V_N^{[N]}=0\) and iterate \(V_j^{[N]}=\mathcal F_j(V_{j+1}^{[N]})\) inward. At a fixed \(j\), two terminal choices at \(N\) differ by at most \(C\kappa_0^{N-j}\), uniformly in every control. Thus \(V_j^{[N]}\) converges uniformly as \(N\to\infty\) to \(V_j\), and \[V_j=\mathcal F_j(V_{j+1}),\qquad \|V_j\|\leq C\sum_{m\geq0}\kappa_0^m\delta_{j+m} \leq C'\delta_j.\] The last inequality uses that \(\delta_j\) decreases. The same contraction proves uniqueness among graph families in the chosen ball. The fixed graph equation says exactly that \[ T_j(\iota_{j+1}c+V_{j+1}c) =\iota_jM_j(V_{j+1})c+V_jM_j(V_{j+1})c . \tag{51}\] Invertibility of \(M_j(V_{j+1})\) proves the graph isomorphism. Moreover \(T_{LQ}V_{j+1}=o(1)O(\delta_{j+1})=o(\delta_j)\); inversion of the finite low matrix yields (50).

We give the continuity argument because it uses less than operator-norm continuity of \(T_j\) on all of \(L^2\). For fixed \(\ell\), the metric on the compact ball of radius \(e^{\ell+1}\) depends on only the finitely many controls whose bumps meet that ball. For fixed smooth boundary data, choose one smooth extension into the ball. For a fixed \(0<\alpha<1\), let \(X=\{v\in C^{2,\alpha}(\overline B):v|_{\partial B}=0\}\). The remaining zero-boundary problem is an equation for the Dirichlet operator \(\mathcal L_z:X\to C^\alpha(\overline B)\), which is bijective by the classical Dirichlet theorem [8]. Its inverse is bounded by the bounded inverse theorem. Converging controls make these operators converge in norm between the indicated Hölder spaces. A Neumann series and the inverse-difference identity then give convergence of the solutions and of their inner traces. Approximation by smooth data and the contraction bound gives \[T_\ell(z_n)v\longrightarrow T_\ell(z)v \quad\text{in }L^2 \quad\text{for every fixed }v\in L^2.\] This is strong continuity. Suppose now that finite-rank graph maps \(V_n:\mathbb R^p\to Q_{\ell+1}\) converge to \(V\) in operator norm. For each coordinate basis vector \(\mathbf e\), \[\begin{align*} \|T_\ell(z_n)V_n\mathbf e-T_\ell(z)V\mathbf e\|_2 &\leq \|T_\ell(z_n)(V_n\mathbf e-V\mathbf e)\|_2 +\|(T_\ell(z_n)-T_\ell(z))V\mathbf e\|_2\\ &\leq\|V_n\mathbf e-V\mathbf e\|_2 +\|(T_\ell(z_n)-T_\ell(z))V\mathbf e\|_2 \longrightarrow0. \end{align*}\] The same statement holds on the fixed low basis vectors \(\iota_{\ell+1}\mathbf e\). A finite basis therefore gives operator-norm continuity for precisely the maps from \(\mathbb R^p\) that enter \(\mathcal F_\ell\). Their low inverses are finite matrices with the uniform inverse bound already proved, so inversion is continuous. Induction shows that each finite inward graph iterate is continuous in the finitely many controls on its finitely many balls. Its error from \(V_j\) is at most \(C\kappa_0^{N-j}\), uniformly over the complete control product. Given an error tolerance, first choose \(N\) for this tail and then require closeness of those finitely many controls. This proves product-topology continuity of \(V_j\). The same finite-rank estimate and finite-matrix inversion prove continuity of \(A_j\). In particular the argument includes the dependence on earlier controls and the graph’s dependence on arbitrarily late controls; it asserts no unnecessary norm continuity for \(T_j\) on the infinite-dimensional space. ◻

Proof. Fix such a mesh point and temporarily write \(\beta=b(j)^{-2}\) and \(\epsilon=z_\nu\eta_\nu^{3/4}\). On the plateau the only active perturbation is \(L_\epsilon=L(w(s(j))+\epsilon\psi_\nu)\). Put \[q_\beta(x)=e^{\mathfrak d(\beta x)},\qquad p_\beta(x)=q_\beta(x)^{-1}.\] Let \(\Pi_\epsilon\) be the isolated first-\(p\) projection and \(\Pi=\Pi_0\). The compression to \(E=\Pi L^2\) of the frozen inward operator is \(C_-(\epsilon)=\Pi p_\beta(L_\epsilon)\Pi|_E\). Its part passing through the high subspace \(I-\Pi_\epsilon\) is \(O(\epsilon^2)\): the input and output projections each have norm \(O(\epsilon)\), while \(p_\beta(L_\epsilon)\) is a contraction. The remaining part is a smooth finite-cluster matrix. Differentiating it in the reference eigenbasis gives \[(C_-'(0))_{hi} =p_\beta[\lambda_h(s(j)),\lambda_i(s(j))] \langle e_h(s(j)),\dot L_\nu(s(j))e_i(s(j))\rangle_2 ,\] where \(q[x,y]=(q(x)-q(y))/(x-y)\) for \(x\neq y\), with the continuous extension \(q[x,x]=q'(x)\), and the same notation is used for \(p_\beta\). This is the derivative of the entire isolated cluster matrix, including at a double eigenvalue.

At \(\epsilon=0\), \(C_-(0)\) is diagonal with entries \(p_\beta(\lambda_i)\). Differentiating its inverse and using \[-q_\beta(x)q_\beta(y)p_\beta[x,y]=q_\beta[x,y]\] shows that the off-diagonal derivative of \(C_-(\epsilon)^{-1}\) is \[q_\beta[\lambda_h(s(j)),\lambda_i(s(j))] \langle e_h(s(j)),\dot L_\nu(s(j))e_i(s(j))\rangle_2 .\] This defines \(m_{hi}(s(j),b(j))\). At equality of the eigenvalues, \[q_\beta'(\lambda) =e^{\mathfrak d(\beta\lambda)} \beta\,\mathfrak d'(\beta\lambda),\] which proves (53). The factor \(\beta\) occurs exactly once, from differentiating \(q_\beta\); there is no extra \(\beta\) in front of its divided difference. The divided difference is continuous across equality. The nonzero angular entry therefore permits a fixed neighborhood on which its product has one sign and is bounded away from zero, uniformly for \(b\) near \(a\). Self-adjointness of \(\dot L_\nu\) in the fixed real Hilbert space makes the reversed first-order entry equal at the crossing.

The finite-cluster Taylor remainder is \(O(\epsilon^2)\), uniformly in the fixed phase neighborhood. Replacing the frame at \(j+1\) by that at \(j\) costs \(O(\eta(j))\). Lemma 10 replaces the frozen inward map on this frame by \(T_j\) with \(o(\delta_j)\) error, and Proposition 11 contributes only \(T_{LQ}V_{j+1}=o(\delta_j)\). The inverses involved are finite matrices with a uniform bound depending on the fixed \(\tau_*>0\); hence these errors remain of the same order after inversion. A fixed phase neighborhood has time width \(O(\eta_\nu^{-1})=o(t_\nu)\), so \(\eta(j)/\eta_\nu\to1\) uniformly there as the start tends to infinity. Thus \(O(\epsilon^2)=O(\delta_j^2)=o(\delta_j)\), and the stated expansion follows. The remainders are uniform in every other control because the frozen calculation is local, the frame comparison is uniform over whole-ball histories, and the graph estimate is uniform over all future tails. ◻

Proof. For a fixed ordered pair define the reference logarithmic ratio at phase \(s\) and radial factor \(b\) by \[\ell_{hi}(s,b) =\mathfrak d(b^{-2}\lambda_h(s)) -\mathfrak d(b^{-2}\lambda_i(s)).\] On a sign interval \([s_a,s_b]\) bounded by successive zeros, simple splitting and compactness of the finitely repeated types give \[ |\ell_{hi}(s,b)| \geq c_*\min\{s-s_a,s_b-s,1\}, \tag{70}\] uniformly for \(b\) near \(a\). Near either endpoint there is also the upper bound \(C\) times the distance to that endpoint. The lower bound follows near a zero from its nonzero phase derivative and away from all zeros from compactness. The common radial factor does not alter the zeros, and \(\mathfrak d'\) has positive upper and lower bounds on the finite spectral range. The phase lengths of these intervals have positive lower and finite upper bounds. Initial intervals are merely truncated subsets of a sign interval for the periodic reference data, so the same estimates apply to them after periodically extending the reference data for this comparison.

Over a bounded phase interval, the elapsed time is \(O(t^{3/4})=o(t)\). Thus the speed at every point of one such interval is comparable to a single value \(\eta_*\), with relative error tending to zero as \(J\) increases. Consecutive mesh phases are separated by comparable multiples of \(\eta_*\). Consider a stable product over \(m\) consecutive steps inside the interval. Apart from a bounded number of endpoint rounding steps, at least a fixed fraction of those phases have distance at least \(c_*\eta_*m\) from both endpoints. To see this, phases within distance \(d\) of the two endpoints number at most \(C(d/\eta_*+1)\); choose \(d=c_*'\eta_*m\) with \(c_*'\) small. Since \(\eta_*m\) is bounded by the fixed maximum phase length, \(\min\{d,1\}\geq c_*''\eta_*m\). Summing (70) along the product therefore gives \[\sum |\log r_{hi}|\geq c_*\eta_*m^2-C.\] The sign in a stable direction makes the product the exponential of the negative of this sum. The one-step ceiling convention, including a possible neutral step, only changes the constant. Every stable weight therefore obeys \[ W(m)\leq C e^{-c_*\eta_*m^2},\qquad \sum_{m\geq0}W(m)\leq C\eta_*^{-1/2}. \tag{71}\] The second estimate follows by comparison with the Gaussian integral.

On a switching interval, \(\delta_q\) is comparable to \(\eta_*^{3/4}\). Combining (62) with the Green sum in (71) bounds every stable trial propagation by \(C\eta_*^{3/4}\eta_*^{-1/2}=C\eta_*^{1/4}\), which is comparable to \(\eta(j)^{1/4}\) at the output step. For a noncrossing pair compactness instead gives a fixed \(\rho<1\) with either \(r_{hi}\leq\rho\) or \(r_{hi}^{-1}\leq\rho\) at every step. The backward tail is bounded by \(\sum_{q=j}^{\infty}C\rho^{q-j}\delta_q\leq C'\delta_j\), because \(\delta_q\) decreases. For the forward sum, whose initial interval can be arbitrarily long, use \[\frac{\delta_q}{\delta_j}=(j/q)^{9/16} \leq(1+j-q)^{9/16}\qquad(J\leq q\leq j).\] It follows that \[\sum_{q=J}^{j-1}\rho^{j-1-q}\delta_q \leq\delta_j\sum_{m\geq0}\rho^m(m+2)^{9/16} \leq C'\delta_j.\] This is also bounded by \(C\eta(j)^{1/4}\). Finally \(C\eta^{1/4}<\eta^{1/8}\) once \(J\) is late, proving (68).

For the sign at the crossing \(\nu\), take a fixed small \(\theta>0\) and the mesh window \[|j-t_\nu|\leq\theta\eta_\nu^{-1/2}.\] Its phase width is \(O(\theta\eta_\nu^{1/2})\), so the window lies inside the fixed plateau for late \(J\). The upper linear bound on \(|\ell_{hi}|\) near the zero shows that an incoming weight from any index in this window is at least \(\exp(-C\theta^2)\): the sum of its logarithmic losses is at most \(C\eta_\nu\sum_{m\leq \theta\eta_\nu^{-1/2}}(m+1)\). The two incoming sides together contain at least \(c_*\eta_\nu^{-1/2}\) such indices. At \(z_\nu=\pm1\), all their forces have the sign in (63) and magnitude at least \(c_1\eta_\nu^{3/4}\). Their contribution to (67) has that sign and magnitude at least \(c_2\eta_\nu^{1/4}\).

Every other term on the plateau has the same sign. Outside the fixed plateau the phase distance to the crossing is bounded below by a positive constant. Its incoming weight contains a stable segment of length \(c_*\eta_\nu^{-1}\); (71) bounds the weight by \(C e^{-c_*'/\eta_\nu}\). The two adjacent switching intervals contain at most \(C\eta_\nu^{-1}\) indices, and their forces are \(O(\eta_\nu^{3/4})\). Their total contribution outside the plateau is therefore at most \[C\eta_\nu^{-1/4}e^{-c_*'/\eta_\nu} =o(\eta_\nu^{1/4}).\] For a truncated initial interval, separation of the start from the first crossing places the central window after \(J\) for late starts. Deleting earlier terms does not change its signed lower bound, and the retained outside-plateau terms satisfy the same tail estimate. The sign from the central window dominates. Reverse the orientation of \(z_\nu\) when needed so that its positive endpoint gives positive mismatch. Uniformity of (63) over every other variable yields (69). ◻

Completion of Theorem 1. Given \(4/9<v<1\) and \(1<c<9v/4\), set \(a=\sqrt v\), choose \(\sqrt c<\vartheta<3a/2\), and then choose \(a^2<\kappa_*<1\). Fix the equatorial bump and a sufficiently small convex near-round neighborhood. It may be made arbitrarily small before \(k\) is chosen; in particular, impose the strict frequency bound (24). Lemma 7 and Proposition 8 impose only cofinite conditions on the integer \(k\). Choose one threshold above all of them and large enough that \((\lfloor\vartheta k\rfloor+1)^2\geq c(k+1)^2\). For each integer above this threshold choose its finite program, crossing directions, gaps, and disjoint sign neighborhoods. Choose the Ricci buffer \(C\) next. Proposition 9 works for every sufficiently large \(J\) after these choices. Enlarge \(J\) to meet the freezing-slab, relative graph, box, and endpoint-sign requirements; none imposes an upper bound on \(J\).

The simultaneous fixed point selects one control sequence for all \(p\) lines, and the construction above gives \(p\geq c(k+1)^2\) independent entire functions in \(\mathcal H_k\) for its metric \(g_k\). The radial proposition gives smoothness, completeness, the Euclidean core, nonnegative Ricci curvature, and positive Ricci curvature outside a compact set. Section 4.2 gives \(\operatorname{AVR}(g_k)=a^2=v\), uncountably many nonisometric pointed cone limits, and the negative radial planes along a sequence tending to infinity. These conclusions hold for every provisional control and hence for the selected one. This proves the stated order of quantifiers, with one threshold followed by each eligible \(k\) and then its metric \(g_k\). ◻

Proof of Corollary 2. Fix \(\epsilon>0\) and \(1<c<9/4\). Choose \[\max\{2\sqrt c/3,(1+\epsilon)^{-1}\}<a<1.\] Choose \(0<\delta<1\) small enough that \[a^2(1-\delta)\geq(1+\epsilon)^{-2}, \qquad 1+\delta\leq(1+\epsilon)^2.\] Run the preceding construction with \(v=a^2\), taking the initial conformal neighborhood small enough for (6) with this \(\delta\). The reference program and its interpolation are chosen with a positive margin inside that neighborhood, so all controls remain there once \(J\) is sufficiently large. The global comparison (39) gives the asserted tensor bounds. Integrating along curves and taking infima gives the stated distance bounds. The choice of \(a\) can be arbitrarily close to \(1\), so the volume ratio can be prescribed arbitrarily close to \(1\) as well. ◻