A counterexample to integer-degree harmonic dimension comparison

Proof. We suppress J from the notation. Relative to g_0, the metric \widetilde g_q has eigenvalues 1 on the (m-1)-dimensional horizontal space and q on the Hopf direction. Its volume form is \sqrt q\,d\operatorname{vol}_{g_0}. Multiplication of the metric by a^2q^{-1/m} multiplies this volume form by a^m q^{-1/2}, proving the volume assertion.

Here is a direct curvature calculation. Let \nabla be the round connection and set \phi X=\nabla_X\xi. Tangent projection of the ambient derivative gives

and differentiation gives

Put \delta=q-1, which is constant on the sphere. If C=\widetilde\nabla-\nabla is the difference between the connections of \widetilde g_q and g_0, the Koszul formula yields

The vectors \phi X,\phi Y are horizontal, so

In particular, \tr[X\mapsto C(X,Z)]=0. Tracing the connection change formula for curvature therefore gives

These are endomorphism traces and can be computed in a round orthonormal frame. Using the identities for \phi, the four terms obtained by differentiating C contribute, in order,

Thus the first trace is \delta(-2g_0+2m\eta^2)(Y,Z). For the second trace, one has

whose trace in X is -(m-1)\delta^2\eta(Y)\eta(Z). Since \Ric_{g_0}=(m-1)g_0, this proves the tensor identity

The squared length of \xi in \widetilde g_q is q. Hence the vertical eigenvalue of its Ricci endomorphism is (m-1)q, while the horizontal eigenvalue is (m-1)-2(q-1). Constant metric scaling leaves the covariant Ricci tensor unchanged and divides its endomorphism eigenvalues by the scale. This proves the stated eigenvalues and (2.1).

Finally, the inverse metric is

Because its volume form is a spatially constant multiple of round volume, its divergence on vector fields is the round divergence. The field \xi has zero round divergence, so \operatorname{div}_{g_0}((D_Ju)\xi)=D_J^2u. It follows that

Restricting this identity to V_l gives (2.2).□

Proof. Complexify V_l and decompose homogeneous polynomials into bidegrees (b,l-b) in z_1,\ldots,z_s and their conjugates. The operator D_{J_0} acts on this bidegree by \mathrm i(2b-l). The contraction \sum_j\partial_{z_j}\partial_{\bar z_j} maps onto the polynomials of bidegree (b-1,l-b-1): in the positive monomial inner product in which z^\alpha\bar z^\beta has squared norm \alpha!\beta!, its adjoint is multiplication by \sum_j z_j\bar z_j, which is injective. Taking the difference of dimensions gives the first line of (3.2); cancellation gives the second. The contributions indexed by b and l-b belong to the same eigenvalue of -D_{J_0}^2 and are added; the middle contribution is counted once. Since this operator is real symmetric, the complex dimension of each complexified eigenspace equals the real dimension of the corresponding original real eigenspace. There is no additional factor of two.

Summing the second line by the binomial convolution identity gives

More precisely, h_{l,b}/N_l is a beta-binomial probability. If X has density proportional to [x(1-x)]^{s-2} on (0,1) and, conditionally on X, K_l is binomial with parameters l,X, then integration of the binomial probability against this density gives \mathbb P(K_l=b)=h_{l,b}/N_l. Since

we have K_l/l\to X in probability in this representation. The change of variable y=(2x-1)^2 gives density proportional to y^{-1/2}(1-y)^{s-2}, proving the assertion.□

Proof. Put r=m-1. The beta density, equivalently polar coordinates for independent standard normal variables, gives the representation

where Z is standard normal and W_r is an independent sum of r squared standard normal variables. Since \mathbb E[W_r/r]=1 and \operatorname{Var}(W_r/r)=2/r, we obtain rY_m\Rightarrow Z^2. The limiting distribution is continuous and strictly increasing on (0,\infty), so its upper quantile y_\infty(v) satisfies r y_m(v)\to y_\infty(v) for every 0<v<1. Fatou’s lemma therefore gives

where R=|Z| and Q(x)=\mathbb P(R>x). The last equality follows since Q(R) is uniform on (0,1). This expectation is finite: integration of the half-normal density on [x,x+1] gives Q(x)\ge\sqrt{2/\pi}\,e^{-(x+1)^2/2}.

For x>0, integration of the same density gives the classical Mills upper bound (see [12, Section 4, p. 113])

Consequently, writing A=\mathbb E[R^2\log R], we have

Integration by parts against the half-normal density yields \mathbb E[R^4\log R]=3A+1. Under the probability measure d\nu=R^4\,d\mathbb P/3, Jensen’s inequality for the logarithm applied to 1/R now gives

Combining this with (3.6) proves

Here the final strict inequality can be checked without decimal approximations. Indeed, \arctan x>x-x^3/3 for x>0 gives

so 3\pi/4>7/3. Strict convexity of t\mapsto1/t gives the midpoint bounds

Equations (3.5) and (3.7) imply (3.4).□

Proof. Choosing the link. Choose an odd m\ge7 satisfying Lemma 3.2, and fix

For a small positive parameter \varepsilon, set

For q\ge1, the right-hand side of (2.1) is

At q=q_*, the required strict inequality follows from c>2/(m-1); it consequently holds throughout [1,q_*]. In particular a_*^2<F(q_*)<1.

Limiting means. By (2.2), the eigenvalues at the dwell parameters are the numbers

with the multiplicities in Lemma 3.1. Thus their increasing order corresponds to the decreasing order of the Hopf weights (2b-l)^2. If d(d+m-1)=\lambda and d\ge0, then 0\le\sqrt\lambda-d\le(m-1)/2. Since B_l/l^2\to1, Lemma 3.1 and convergence of quantiles to a continuous, strictly increasing distribution imply

These rescaled exponents are uniformly bounded, so averaging the first \lceil uN_l\rceil terms gives, for every 0<u<1,

One may see this directly by integrating the empirical quantile in (3.11); the possible last fractional term has size O(N_l^{-1}) after division by l.

Define

All functions here are bounded away from zero for sufficiently small \varepsilon. Uniformly for 0<v<1,

The same uniform remainder remains valid after averaging over (0,u). Expansion of g_\varepsilon(u)^{-m} and integration therefore give

The second equality uses the nonnegative integral identity \int_v^1du/u=\log(1/v). Fix such an \varepsilon, as small as desired, from now on.

Counting at integer thresholds. For each positive integer k and l\ge2, let M_l(k) be the largest initial group of the ordered exponents whose mean is strictly less than k, taking M_l(k)=0 if no group qualifies. Only finitely many degrees contribute: the displayed eigenvalue formula bounds all \lambda_{l,i} below by a positive constant times l^2, so \min_i d_{l,i} grows at least linearly in l. The initial-group means are nondecreasing, and hence

For fixed u\in(0,1) and \delta>0, (3.12) bounds the mean in this indicator by l(g_\varepsilon(u)+\delta/2) for every sufficiently large l. It is therefore strictly below k whenever l\le k/(g_\varepsilon(u)+\delta), including at the upper endpoint. Using (3.3), the sum of the corresponding N_l, divided by \sum_{l=0}^kN_l, has limit (g_\varepsilon(u)+\delta)^{-m} as the integer k\to\infty. Letting \delta\downarrow0 and applying Fatou’s lemma to (3.14) yields

Discarding the finitely many lower degrees in this argument changes no limit. Consequently every sufficiently large integer k has a strict surplus. Taking L to be the largest degree with M_l(k)>0 and setting M_l=M_l(k) proves (3.9).□

Proof. We first prove the assertion in a single degree l. The nonzero factor \kappa_l has no effect on this assertion. Let \mathfrak a\subset\mathfrak{sl}(V_l) be the real Lie algebra generated by (D_J^2)_{\mathrm{tf}}, and set

Transpose on W will always refer to the complex bilinear extension of the real round inner product. The single-degree argument first produces a rank-one symmetric endomorphism of W. Round conjugation and brackets then produce all trace-free endomorphisms of W. Finally, we separate the different degrees to obtain simultaneous control on their direct sum.

Diagonal operators and individual weight blocks. Choose orthonormal coordinates (x_1,y_1,\ldots,x_s,y_s) and write z_i=x_i+\mathrm i y_i. For \theta\in\mathbb R^s, let R_\theta rotate coordinate plane i by \theta_i, and let T_\theta u(x)=u(R_\theta x). The weight space of \nu\in\mathbb Z^s is the character eigenspace

Thus z_i has weight e_i, \bar z_i has weight -e_i, and D_{J_0} acts on W_\nu by \mathrm i\sum_i\nu_i. In particular, a monomial of holomorphic degree b and antiholomorphic degree l-b has Hopf charge 2b-l, consistently with Section 3. Every occurring weight satisfies

Every weight on the extreme layer

occurs and has a one-dimensional weight space: it is spanned by the monomial using z_i^{\nu_i} when \nu_i\geq0 and \bar z_i^{-\nu_i} when \nu_i<0. This monomial is harmonic, since in each coordinate plane it involves at most one of z_i,\bar z_i. It is the only degree-l monomial of that weight.

More generally, every weight satisfying the displayed bound and parity condition occurs. To check its multiplicity, set a=(l-\sum_i|\nu_i|)/2, a nonnegative integer. Every monomial of weight \nu is the corresponding signed monomial of degree \sum_i|\nu_i|, multiplied by \prod_i(z_i\bar z_i)^{a_i} with a_i\ge0 and \sum_i a_i=a. The full polynomial weight space therefore has dimension \binom{a+s-1}{s-1}. The polynomial Laplacian preserves the weight and maps onto the corresponding weight space two degrees lower: its adjoint in the positive monomial inner product is a nonzero scalar multiple of multiplication by \sum_i z_i\bar z_i, which preserves weight and is injective. Thus the harmonic weight multiplicity is

where the second term is zero when a=0. This standard weight formula also appears in Lauret–Miatello–Rossetti [10, Lemma 3.6]. In particular, interior weight spaces may have dimension greater than one. The interpolation below isolates an entire source-to-target weight block; we use a rank-one conclusion only when both endpoints are on the extreme layer.

Conjugating by a round rotation sends the family of generators to itself. Thus \mathfrak a_{\mathbb C} is invariant under the round torus and contains each torus Fourier component of any of its elements. Indeed, Fourier projection is an integral of scalar multiples of torus conjugates, and a finite-dimensional vector subspace is closed.

For \sigma=(\sigma_1,\ldots,\sigma_s)\in\{\pm1\}^s, let J_\sigma rotate coordinate plane i with sign \sigma_i. On the \nu-weight space, D_{J_\sigma}^2 acts by -(\sum_i\sigma_i\nu_i)^2. For i\ne j, the identity

shows that \mathfrak a_{\mathbb C} contains a diagonal operator A_{ij} acting on weight \nu by \nu_i\nu_j+c_{ij}, where c_{ij} is independent of \nu. The scalar term has no effect on brackets.

An endomorphism Fourier component of weight shift \delta satisfies T_\theta A T_\theta^{-1} =e^{\mathrm i\sum_i\delta_i\theta_i}A; equivalently, it maps each W_\nu into W_{\nu+\delta}, with absent weight spaces understood to be zero. On its block from W_\nu to W_{\nu+\delta}, the operator \operatorname{ad}A_{ij} acts by

If \delta has at least three nonzero coordinates, these joint eigenvalues distinguish all source weights. In fact, equality for two sources, with difference v, implies

On the support of \delta, the ratios v_i/\delta_i have pairwise sums zero; three such coordinates force every ratio to be zero. Pairing any coordinate outside the support with one in the support then gives v_i=0 there as well.

There are only finitely many weights. Polynomials in the commuting operators \operatorname{ad}A_{ij} therefore project onto an individual source block within a fixed shift component. More explicitly, for each competing source choose one coordinate of (4.2) that distinguishes it from the desired source, and multiply the corresponding linear interpolation factors. These operations preserve \mathfrak a_{\mathbb C}. When source and target belong to \Lambda_l, the isolated nonzero block is a rank-one directed matrix.

A rank-one symmetric operator. Regard an extreme weight as l signed tokens, with no opposite signs at the same coordinate. We claim that the algebra contains the directed matrices for every legal move

where the two latter signs are arbitrary and both endpoints must remain in \Lambda_l. Such a shift has three nonzero coordinates. It remains to exhibit a generator with a nonzero block for the move.

Here is an explicit simultaneous choice of the needed coefficients. For three distinct coordinate indices i,j,k, let X_a,Y_a denote their real coordinate vectors and set

with the three reverse images determined by J^2=-I. Use the standard complex structure on the remaining coordinate planes. This defines an orthogonal complex structure. For signed variables u_a^\epsilon=x_a+\mathrm i\epsilon y_a, \epsilon\in\{\pm1\}, its vector field satisfies

For every choice of the three signs, the relevant replacement coefficients are

For completeness, there is no cancellation with other terms of D_J^2. In the independent complex linear coordinates z_a,\bar z_a, the identity D_J^2u=-u for linear functions gives, on homogeneous degree-l polynomials,

If

is an extreme monomial, where M uses the other coordinate planes, the coefficient of the split monomial is \mathrm i\epsilon\,a(a-1)/2, and that of the merge monomial is -\mathrm i\epsilon\,bc/2. They are nonzero whenever the indicated move is legal. Indeed the exponent deficits of the target force the differentiated variables to be precisely the two variables specified by the move. Since rotations commute with the Euclidean Laplacian, D_J^2p is already harmonic; no harmonic projection changes these coefficients. Trace subtraction does not affect this shift. Fourier projection and the block isolation above now give the claimed directed matrices.

We next join -le_1 to le_1 by a simple path of these moves. For even l=2h, use

splitting negative pairs at coordinate 1 in the first half and merging the tokens at 2,3 into positive pairs at 1 in the second half. This includes l=2, for which the path is -2e_1,e_2+e_3,2e_1.

For odd l=2h+1, h\geq1, both starting signs admit the path

The last step merges the remaining token at 1 and one positive token at 2 into two positive tokens at 4. Reverse the positive-start path and append it to the negative-start path. The resulting path is simple: the two branches have opposite nonzero first coordinates until their common endpoint, and no branch repeats a weight. For l=3, that endpoint is e_3+2e_4. All intermediate weights remain extreme.

Choose monomial bases on these one-dimensional weight spaces and normalize their directed matrices to matrix units. Nested brackets along a simple path give its endpoint unit, by [E_{c,b},E_{b,a}]=E_{c,a} for distinct vertices. Hence \mathfrak a_{\mathbb C} contains the unit from weight -le_1 to weight le_1. If v=z_1^l, torus invariance of the bilinear round pairing shows that v pairs only with weight -le_1, and

The endpoint unit is consequently a nonzero multiple of vv^{\mathsf T}. Notice also that v^{\mathsf T}v=0.

From the rank-one operator to all trace-free matrices. All round conjugates of vv^{\mathsf T} belong to the algebra. We first verify that the round orbit of v spans W. Up to a nonzero scalar, its members are the polynomials (a\cdot x)^l with

To see this, write a=b+\mathrm i c with b,c real. The isotropy condition says that b,c are orthogonal and have equal positive length. After rescaling, this ordered pair is the image of the first coordinate pair under a real special orthogonal transformation.

On homogeneous degree-l polynomials use the Fischer bilinear form

Its restriction to real harmonic polynomials is positive definite, so its complex bilinear restriction to W is nondegenerate. If p\in W annihilates the orbit powers, then

shows that p vanishes on the isotropic quadric. Therefore p is divisible by the quadratic polynomial Q(x)=\sum_{a=1}^s(x_a^2+y_a^2). An elementary verification is to divide by this monic polynomial in the last variable y_s; the remainder has the form A(x')y_s+B(x'), where x' denotes the remaining variables. Evaluating at the two distinct roots for generic x' forces both A and B to vanish identically.

Such a harmonic multiple must be zero. Indeed multiplication by Q is adjoint to the Euclidean Laplacian in the positive Hermitian Fischer form. If p=Qq and \Delta p=0, then

Thus no nonzero element annihilates the orbit, proving its span.

For two orbit vectors u,w,

For fixed u\ne0, the factor u^{\mathsf T}w is not identically zero on the orbit, by its spanning property and nondegeneracy of the round pairing. It is a real-analytic function of the round rotation. The real special orthogonal group is connected, so its nonzero locus is dense. Division by this factor and then passage to limits show that \mathfrak a_{\mathbb C} contains uw^{\mathsf T}-wu^{\mathsf T} for every pair of orbit vectors. Their span is the full complex skew-symmetric algebra, since the orbit spans W.

For every real skew-symmetric K, the operator \exp(\operatorname{ad}K) preserves \mathfrak a_{\mathbb C}; this uses only bracket closure in the complex algebra. It follows that \mathfrak a_{\mathbb C} is invariant under conjugation by the full real group SO(V_l). Here one may use that every special orthogonal matrix is a product of plane rotations. This group on the harmonic-function space sends v, up to a scalar, to every nonzero isotropic vector of W: their real and imaginary parts are orthogonal equal-length pairs, and \dim V_l\geq3. Consequently the algebra contains ww^{\mathsf T} for every isotropic w\in W.

These matrices span all trace-free symmetric matrices. In a real orthonormal basis, the rank-one matrices associated with e_i+\mathrm i e_j and e_i-\mathrm i e_j have sum 2(E_{ii}-E_{jj}) and difference 2\mathrm i(E_{ij}+E_{ji}). Together with the skew-symmetric matrices already obtained, they span \mathfrak{sl}(W). We have proved \mathfrak a_{\mathbb C}=\mathfrak{sl}(W). Since \mathfrak a\subset\mathfrak{sl}(V_l) is a real vector subspace and complexification preserves dimension, it follows that \mathfrak a=\mathfrak{sl}(V_l).

Independence of the degrees. Let \mathfrak h be the Lie algebra generated by the tuples (4.1). Every one-factor projection is full by what we have just proved. Moreover

is strictly increasing in l. The real Lie algebras \mathfrak{sl}(V_l) are therefore simple and pairwise nonisomorphic. Simplicity can be seen directly: in any nonzero ideal, a bracket with a matrix unit first produces an element with an off-diagonal entry if necessary; polynomials in the adjoint action of a generic trace-free diagonal matrix isolate an off-diagonal unit. Brackets with the other matrix units then produce all off-diagonal units and all diagonal differences.

We give the resulting Lie-algebra Goursat argument explicitly; for the two-factor formulation, see [7, Section 3]. Inductively, suppose the projection onto a preceding product \mathfrak b is full, and let \mathfrak c be the next simple factor. The projected algebra \mathfrak h'\subset\mathfrak b\oplus\mathfrak c is surjective onto both sides. The kernel of its projection onto \mathfrak b is an ideal of \mathfrak c, and hence is either zero or \mathfrak c. In the latter case the whole product is present. In the former case \mathfrak h' is the graph of a surjective homomorphism \mathfrak b\to\mathfrak c. The image of each simple summand of \mathfrak b is an ideal in \mathfrak c; a nonzero image would make that summand isomorphic to \mathfrak c. This contradicts their distinct dimensions. The induction proves the claimed full direct sum.□

Proof. Let \mathcal H be the subgroup of \mathcal G consisting of finite products of \exp(tX_J) with arbitrary real t. Define

Conjugation by every generator exponential preserves E. Differentiating shows that E is stable under bracketing with each X_J. Since E contains the generators, it contains their generated Lie algebra, which is the whole Lie algebra of \mathcal G by Proposition 4.1. Choose a basis \operatorname{Ad}_{h_a}X_{K_a}, 1\leq a\leq d, from this spanning set. The map

has value I at zero and differential

which is an isomorphism. Every factor h_a, and its inverse, is a fixed finite word of generator exponentials. Thus every value of F is in \mathcal H.

The inverse function theorem gives an identity neighborhood contained in \mathcal H, making \mathcal H an open subgroup. The group SL_N(\mathbb R) is connected: polar decomposition reduces this to connectedness of SO(N) and of the positive definite determinant-one matrices, the latter being joined to the identity by S^t, 0\leq t\leq1. Hence \mathcal G is connected, and its open subgroup \mathcal H is all of \mathcal G.

Choose a fixed finite word representing the desired target and multiply it by F. The resulting word has that target as its value at zero and still has invertible differential. Expanding the fixed conjugating words gives the asserted form, with fixed complex structures and smooth real time parameters.□

Proof. The radial scale. Use the fixed smooth nondecreasing cutoff \chi from Section 3. For t>2, put

and set b=0, a=1 for t\leq2. The integral of \chi(t)(1+t)^{-5/4} over (2,\infty) is positive and finite. Thus there is a unique \mu>0 for which a(t)\to a_*; moreover \mu\to0 as a_*\to1. Condition (3.8) gives b\geq-1/4. Since \chi'\geq0, for t>2 we have

The last factor is positive for t>2 and b\geq-1/4. The tail identities (5.2) follow by integration.

The slice shape and the mixed Ricci tensor. Write g_r=r^2\gamma(\log r) for the metric on a slice. A dot will denote differentiation in t=\log r. Identify the tangent bundles of the slices using the product coordinates. Their shape endomorphism is

Where a nonround part of a bump or a ramp occurs, J is fixed. Direct differentiation of G gives

On round gaps H=0. Thus the same identities apply everywhere, without requiring derivatives of the labels J(t).

The projection P_J is also the orthogonal projection for \gamma onto the Hopf direction. Indeed, \xi_J has constant squared length a^2q^{1-1/m} on a fixed slice, and its metric dual is the same constant times \eta_J. The flow of \xi_J preserves both g_0 and \eta_J, so \xi_J is a Killing field for \gamma. A Killing field of constant length has zero divergence and \nabla^\gamma_{\xi_J}\xi_J=0: the latter identity follows by pairing the Killing equation with \xi_J. Equivalently, its unit normalization E satisfies

Therefore \operatorname{div}_\gamma P_J=0, and \operatorname{div}_\gamma H=0, because the coefficients in (5.3) are constant on a slice. The trace \tr S=r^{-1}m(1+b) is also spatially constant. The Codazzi formula consequently makes the mixed Ricci tensor identically zero.

We record the remaining curvature identities explicitly. For a metric dr^2+g_r, the connection components involving the radial variable are \Gamma^r_{ij}=-(g_r)_{ik}S^k{}_j and \Gamma^i_{rj}=S^i{}_j. They give the normal curvature endomorphism -\partial_rS-S^2. Its trace is \Ric_g(\partial_r,\partial_r). Tracing the tangential Gauss equation contributes S^2-(\tr S)S to the intrinsic slice Ricci endomorphism. Adding the normal contribution gives

Here an index of the tangential tensor is raised using g_r. The mixed formula is \operatorname{div}_{g_r}S-d(\tr S), already shown to vanish. Substitution of S=r^{-1}(\Id+B), and \tr B=mb, yields

In particular, the H^2 terms cancel from the tangential identity.

Uniform positivity on the periods. All constants in the estimates that follow may depend on the fixed pulse data and K, but not on j or on any choices of h_j. On a pulse,

On either ramp, |\dot q|=O(T_j^{-1}) and |\ddot q|=O(T_j^{-2}). There is no variation of q during the dwell or rest. Taking j_0 large makes q>0 throughout all pulses, including those with negative amplitudes. The metrics on all slices then lie in a fixed uniformly equivalent family. Formula (5.3) gives, throughout period j,

The stronger pulse bound is \tr(H^2)=O(T_j^{-4}). To obtain the derivative estimate, note that P_J is independent of t where q varies; on every label-changing gap H vanishes identically.

From (5.1),

On the scale tail, an exact calculation gives

It has a positive lower bound of order j^{-10}, uniformly on period j, for all sufficiently large j. This dominates the O(j^{-12}) term \tr(H^2). The radial component in (5.4) is therefore positive once j_0 is large enough.

For the tangential component, the strict inequality (2.1) on the compact interval [1,q_*] implies that, for some fixed \delta>0,

All J give the same intrinsic Ricci eigenvalues. Since a(t)\to a_*, this estimate persists with 2\delta replaced by \delta along sufficiently late ramps, dwells, and round rests. The pulse values satisfy q\to1 uniformly; the strict inequality at (a_*,1) likewise supplies the same conclusion on late pulses, after decreasing \delta if necessary. This argument allows q<1 on a pulse. Thus all sufficiently late slices satisfy

On the other hand, B=b\Id+H, b,\dot b\to0, and the displayed bounds for H,\dot H show uniformly that

The tangential component of (5.4) is consequently positive as well. All the thresholds used here depend only on the fixed data and K. Since the mixed component is zero, these estimates prove \Ric_g\geq0 on every period for every admitted parameter sequence.

The initial region and the global manifold. Before t_{j_0} the metric is round in the angular variable:

The scale construction gives

Its radial and tangential sectional curvatures are respectively -f''/f and (1-(f')^2)/f^2, as follows from the same normal curvature and Gauss formulas. They are nonnegative, proving the required Ricci condition in this region.

The pulse bumps and ramp profiles are constant or flat at all junctions, and label changes occur only in round open intervals. Thus \gamma, and hence g, is smooth for r>0. Only finitely many periods meet any bounded interval of t, so the infinite sequence of periods introduces no finite accumulation of junctions. For t\leq2, equivalently r\leq e^2, both a and q equal 1. The metric is exactly Euclidean there and extends smoothly across the origin. The volume formula follows from Lemma 2.1.

A radial path from the origin to a point of radius r has length r. Conversely, the length of any such path is at least the total variation of its radial coordinate, and hence at least r. Thus d_g(0,x)=r(x). The closed centered balls are the compact sets \{r\leq R\} in the underlying \R^{m+1}, so the metric is complete. The underlying manifold is connected, smooth, and without boundary.□

Proof. The Frobenius inner product gives

The corresponding upper differential inequality for \|E\| holds also at its zeros, by continuity (or by first replacing the norm by (\|E\|^2+\varepsilon^2)^{1/2} and letting \varepsilon\downarrow0). Multiplication by e^{\gamma t} and integration gives (6.5). Integrating that estimate in t and interchanging the nonnegative integrals gives (6.6).□

Proof. The matrices A_l(t) are symmetric. For sufficiently large allowed j_0, the values of a and q lie in fixed compact positive intervals, and only finitely many V_l are involved. Positivity of (2.2) therefore gives constants

The radial condition b\geq-1/4, together with b\leq0, gives

Choose c_P>0 below all the initial values l so that \lambda_--p_+c_P-c_P^2>0, and choose C_P above all of them so that \lambda_+-p_-C_P-C_P^2<0. We may decrease c_P and increase C_P so that every \theta_l also lies strictly between them.

The Riccati equation preserves symmetry. At a first contact with the lower barrier, a unit vector v with P_lv=c_Pv satisfies

At a contact with the upper barrier the corresponding derivative is at most \lambda_+-p_-C_P-C_P^2<0. These strict inward derivatives prevent an eigenvalue from crossing either barrier. The bounds also keep P_l in a compact set of matrices on every finite time interval, so local existence for the ordinary differential equation continues for all later times. Uniqueness follows from the same local ordinary differential equation theorem. In the Euclidean region, A_l=B_lI and p=m-1, so P_l=lI indeed satisfies the equation there.

Let \mathcal F_l(t) be the fundamental matrix of Y'=P_lY with \mathcal F_l(2)=I. Its determinant is

Moreover,

so its columns solve (6.7). On t\leq2 they are e^{l(t-2)} times constant vectors. Conversely, the two scalar radial factors for (6.7) in this region are e^{lt} and e^{-(l+m-1)t}. Smoothness at the origin eliminates the second factor. Thus every center-regular solution is one of the solutions Y'=P_lY. Invertibility of \mathcal F_l(t) gives the assertion about arbitrary value data.

It remains to prove the reset estimate. The schedule gives

Times on period j and on the final rest immediately preceding it are therefore comparable to j^8. From (5.2),

uniformly on these intervals. On a round interval set E_0=P_l-\theta_l I. Subtracting (6.1) from (6.8) gives the exact difference equation

Here f_l^{\mathrm{rd}}=O(j^{-2}) by (6.12). The symmetric coefficients pI+P_l and \theta_l I have a fixed positive sum of lower eigenvalue bounds. Lemma 6.1, on a final rest of length T_{j-1}, gives

The discrepancy at the start of the rest is bounded by (6.9), independently of the history. For the first pulse, the interval up to t_{j_0} is entirely round. Apply the same estimate on [t_{j_0}/2,t_{j_0}] once j_0 is large. Its length tends to infinity and all its times are comparable to j_0^8, so the same bound follows. This proves (6.10) with the claimed uniformity.□

Proof. The scalar barriers used in Lemma 6.2 also keep v_l between fixed positive constants. Subtracting (6.1) from (6.13) gives

Its initial value is zero. The forcing is O(j^{-2}) by (6.12), and the damping coefficient is uniformly positive. Thus

We retain v_l exactly in the value equation: even the small pointwise difference in (6.15) can have a large integral over T_j.

For the angular operator, expand (2.2) at q=1. Its first derivative in q is

In particular the trace-free part has the positive coefficient a^{-2}(D_J^2)_{\mathrm{tf}} that appears in (4.1). Since q=1+z/T on the pulse and a(t)-a_*=O(j^{-2}), Taylor’s formula gives

At most one bump is nonzero at any time. In the gaps the angular operator is exactly scalar. Smooth boundedness of the profiles and their first h-derivatives proves the uniform remainder bound, also across those gaps.

Set

It is symmetric, and it has a fixed positive lower eigenvalue bound when j is large. Its Riccati residual is

The scalar terms in this expression cancel by (6.13). Using \partial_t(R_l/T)=\partial_\tau R_l/T^2, the remaining identity is

Equations (6.15) and (6.16) therefore imply

Let E_l=P_l-Q_l. No matrices are commuted in the exact difference equation

The lower eigenvalue bounds for P_l,Q_l, and p allow Lemma 6.1 with a fixed damping rate. The profiles vanish near \tau=0, so

by (6.10). From (6.5), (6.6), and (6.17), we obtain

The initial discrepancy contributes its size to this integral, because it is damped; it does not contribute its size times T.

We next control the derivative in a current parameter. Derivatives are taken with the earlier history fixed. Write E_{l,h}=D_hE_l and Q_{l,h}=D_hQ_l; the calculation applies to every unit direction in the finite parameter space. The coefficients of the Riccati equation depend smoothly on h on the pulse, so the ordinary differential equation has smooth parameter dependence there; the uniform barriers keep its solutions inside one fixed bounded set. Differentiating (6.18) and using D_hP_l=E_{l,h}+Q_{l,h} gives the exact regrouping

Its initial value is zero: the past is fixed and R_l(0,h)=D_hR_l(0,h)=0. Also \|Q_{l,h}\|\leq C/T. Apply Lemma 6.1 to (6.20). Its forcing has pointwise norm at most C\eta_j/T, by (6.17) and the first bound in (6.19). Its integrated norm is at most

Consequently

This argument supplies the derivative estimate directly, without an a priori estimate for D_hP_l.

To pass from coefficient errors to value transmissions, let \mathcal B_l(t,h) be the pulse fundamental matrix up to time t, with value I at t_j, and set

Then

The integrated norms of R_l/T and D_hR_l/T are uniformly bounded; those of E_l and E_{l,h} are O(\eta_j). The elementary integral inequality u(t)\leq a+\int_{t_j}^t b(w)u(w)\,dw, b\geq0, implies u(t)\leq a\exp(\int_{t_j}^t b): the right-hand side of the first inequality has derivative at most b times itself. Applying this inequality to the fundamental-matrix equations and their parameter derivatives bounds F_l,G_l,D_hF_l,D_hG_l uniformly.

For completeness, Z_l=F_l-G_l satisfies

The integral bound for E_l gives \sup_t\|Z_l\|\leq C\eta_j. Differentiating this equation introduces the additional terms

Their integrated norms are O(\eta_j), using the preceding uniform bounds and (6.21). The same integral inequality gives \sup_t\|D_hZ_l\|\leq C\eta_j. Evaluation at the pulse end proves the first assertion of (6.14). At that end R_l(1,h)=D_hR_l(1,h)=0. Combining P_l=Q_l+E_l, (6.15), (6.19), and (6.21) proves the second assertion. Every estimate used the earlier history only through the uniform reset and barriers of Lemma 6.2, proving the stated uniformity.□

Proof. Every operator on the leg is a function of I and D_{J_0}^2, and hence is diagonal in the fixed dwell eigenbasis. A scalar coordinate satisfies

Multiplying the equation for y' by the positive integrating factor e^{\int p} gives

Starting with (y,y')=(1,0), positivity of y makes y' nonnegative until any supposed first zero of y; this prevents that zero. Starting with (0,1), the solution is positive for small positive time and the same argument keeps it positive. Thus the final entries of U_{l,j} and V_{l,j} are positive. Formula (6.22) gives positivity of D_{l,j}, and entry by entry

This proves (6.23).

For an incoming value vector y at the start of the period, the data at the pulse end are

Their value at the end of the leg is

Since D_{l,j}=U_{l,j}+\theta_lV_{l,j}, left multiplication by D_{l,j}^{-1} gives (6.24). The order K_{l,j}(P_{l,e}-\theta_lI) is part of this identity; no commutation with P_{l,e} is used. The large matrix D_{l,j} has disappeared from the error multiplier, leaving only the uniform bound (6.23).

To estimate D_{l,j}, take its i-th scalar solution with initial data (1,\theta_l). The positivity just proved allows its logarithmic derivative w=y'/y throughout the leg. It satisfies

The scalar barrier argument in Lemma 6.2, with initial value \theta_l, keeps w between fixed positive constants. During the dwell,

by (6.12) and the fixed value q=q_*. For e=w-d_{l,i}, subtraction gives

The value of e at the dwell entrance is bounded, and its damping coefficient has a fixed positive lower bound. The scalar version of (6.6) yields

The two ramps and the final rest have total duration 3T_j, and their contribution to \int w is O(T_j). Since the solution starts at 1 and ends at (D_{l,j})_{ii}, integration of w=(\log y)' gives the first equality in (6.25). The second follows from T_j/L_j=j^{-1}\to0 and L_j\to\infty. All constants are uniform over the fixed finite list of coordinates.□

Proof. First fix an admitted past and vary the current parameter h. By (6.24),

The bound (6.23) and Lemma 6.3 imply

The estimate is uniform in the preceding history. Each \mathcal T_{l,j} is a value fundamental matrix of Y'=P_lY, so it has positive determinant; so does D_{l,j}. Determinant normalization is therefore defined for their quotient. The matrices H_l(1,h), h\in K, form a compact subset of the positive-determinant matrices. On a neighborhood of this compact set, \mathcal N_l is smooth with bounded derivatives. Normalization is unchanged by the positive scalar e^{-\varphi_{l,j}}, so (6.27) gives

The assertion is first an estimate for the ambient matrix entries and their derivatives, and hence also in any fixed smooth chart containing the compact image under consideration.

We spell out the uniform inversion. Use Euclidean norms on the parameter and coordinate spaces, and the associated operator norms for their derivatives. Choose a coordinate chart \zeta near \Pi in \mathcal G with \zeta(\Pi)=0, and let f=\zeta\circ\Phi and A_0=Df(0). By (6.4), A_0 is invertible. Choose a closed ball \overline B_r centered at 0, contained in the interior of K, whose \Phi-image lies compactly inside the chart and for which

For every sufficiently large j, uniformly in the admitted past, f_j=\zeta\circ\Psi_j is defined on this ball and

This follows from (6.28) and \eta_j\to0. The map

has derivative of norm at most 1/2 on the ball. Also

It is therefore a contraction of the closed ball into itself. Its iterates converge to a fixed point h_j, for which f_j(h_j)=0, or \Psi_j(h_j)=\Pi.

All the preceding estimates were uniform over the past. Choose j_0 once, large enough for them and for the geometric input, after the fixed word and parameter ball have been chosen. The history before j_0 is round. Choose h_{j_0} by the contraction above, and repeat after each finite chosen history. This inductively defines an infinite admitted sequence and hence a single smooth metric of Proposition 5.1. Undoing determinant normalization at each fixed point gives

By (6.27), the determinant of e^{-\varphi_{l,j}}D_{l,j}^{-1}\mathcal T_{l,j} stays bounded above and away from zero on the fixed compact family. Thus

because v_l is uniformly bounded. This proves (6.26). The equality is for the complete period; no residual change of direction is passed to a later period.□

Proof of Theorem 1.1. We convert the exact cycles from Proposition 6.5 into polynomial growth. For each l with M_l>0 and each 1\leq i\leq M_l, choose the center-regular solution with value vector Y(t_{j_0})=e_{l,i}. Lemma 6.2 permits this choice. Near the origin it is r^l times an element of V_l, hence a homogeneous harmonic polynomial in Euclidean coordinates. It is smooth and harmonic there and, by (6.7), on the rest of the manifold.

Let \sigma_l be the cyclic permutation of \{1,\ldots,M_l\} underlying \Pi_l, and define i_{j_0}=i, i_{j+1}=\sigma_l(i_j). Applying (6.26) sends e_{l,i_j} to a signed multiple of e_{l,i_{j+1}}, with absolute multiplier s_{l,j}(D_{l,j})_{i_{j+1},i_{j+1}}. Hence, with

by (6.25), one has

Signs in \Pi_l do not affect this norm.

We verify the weighted average in (7.1). Group the sum of j^7d_{l,i_{j+1}} into complete cycles of length M_l. Within a cycle starting at j, (j+r)^7-j^7=O(j^6) for 0\leq r<M_l, with a constant depending on this fixed length. Since the sum of d_{l,i_{j+r+1}}-\overline d_l in a complete cycle is zero, that cycle contributes only O(j^6) to the difference from the mean-weighted sum. Summing these errors and bounding the at most M_l-1 final terms by O(J^7) gives

Also \sum_{j<J}|\log s_{l,j}|=O(\sum_{j<J}j^6)=O(J^7). The \varepsilon-terms sum to o(J^8): for any positive number, choose a fixed late index after which their absolute values are below that number, and use \sum_{j<J}j^7=O(J^8). The finitely many earlier terms vanish after division by J^8. Finally,

by (5.1). Therefore

The estimate extends to every radius. From Y'=P_lY and (6.9),

The ratio (t_{j+1}-t_j)/t_j tends to zero by (6.11). Thus (7.2) gives

Choose \alpha_l with \overline d_l<\alpha_l<k. For all sufficiently large r=e^t, \|Y(\log r)\|\leq r^{\alpha_l}. The fixed finite-dimensional space V_l has

for example by Cauchy–Schwarz and boundedness of the finitely many smooth basis functions. Since d_g(0,x)=r(x), this gives |u(x)|\leq C_u(1+d_g(0,x))^k outside a compact set. Smoothness bounds u on that compact set and supplies the same inequality everywhere.

These functions are linearly independent: their restrictions to the sphere t=t_{j_0} are the selected vectors of mutually orthogonal round spaces V_l. It follows that

The Euclidean identification was proved at the end of Section 3, so this is the required strict comparison.

Finally, the distance and volume statements in Proposition 5.1 give

Because a(t)\to a_*, changing variables r=Rz and using dominated convergence shows that this quantity divided by \omega_{m+1}R^{m+1} tends to a_*^m; here \operatorname{vol}_{g_0}(S^m)=(m+1)\omega_{m+1}. This completes the proof.□

Proof. Centering a fixed logarithmic window in the round rests or nonround dwells and letting its center tend to infinity gives smooth annular limits with links G(a_*,1,J_0) and G(a_*,q_*,J_0), respectively. They are nonisometric: the first is Einstein, whereas the second has distinct horizontal and vertical Ricci eigenvalues. Every intermediate ramp link also occurs as a limit, because the ramp durations diverge and their derivatives tend to zero. These annular limits extend to pointed cone limits including the vertex: the inner radial ball of radius \delta has diameter at most 2\delta in every rescaling and in the limit. Comparing paths on the complementary annuli, then letting \delta\downarrow0, gives convergence of distances on each bounded ball. All these links have the same volume.

The additional local conformal flatness hypothesis in [2] also fails. Put h_*=G(a_*,q_*,J_0). The nonround cone metric dr^2+r^2h_* is conformal, under t=\log r, to the product dt^2+h_*. For this product the Weyl tensor has mixed component

Indeed the product has zero mixed curvature and radial Ricci tensor; substitution in the trace decomposition of curvature gives this formula. The component is nonzero because the link has distinct horizontal and vertical Ricci eigenvalues. Vanishing of the Weyl tensor is conformally invariant and is preserved by smooth annular limits. If the constructed metric were locally conformally flat, its nonround cone limit would have zero Weyl tensor, contradicting the formula.

For maximal Hopf charge in degree l, its growth root \alpha_l(q) is the nonnegative solution of

This root varies continuously with q. At q=1 its asymptotic slope is a_*^{-1}; at q=q_* it is a_*^{-1}q_*^{-(m-1)/(2m)}, which is strictly smaller. More precisely, the endpoint roots are

The difference of their slopes is positive, so for every sufficiently large l, \alpha_{l+1}(q_*)<\alpha_l(1) and \alpha_l(q_*)<\alpha_l(1). The continuous image of the q-interval contains the entire interval between these endpoints. These intervals overlap and have unbounded upper endpoints, so their union contains a half-line. In particular, equal cone-link volumes do not provide a discrete union spectrum.

The chosen integer k itself lies in this union. Indeed, (3.9) forces some selected degree l>k. Its smallest dwell root is below k, while its round root \theta_l is greater than l because a_*<1. Continuity supplies an intermediate cone root equal to k; the strict endpoint inequalities in fact put a neighborhood of k in \mathcal D(M). Xu’s nonunique-cone three-circles theorem [16, Theorem 3.4] requires the comparison exponent to lie outside \mathcal D(M), so it cannot be applied at k or at any sufficiently nearby exponent. The unique-cone spectral bounds in [8] also have a hypothesis that fails. A global logarithmic growth average, as in (7.2), need not equal the local homogeneous degree seen after normalization on each separate cone subsequence.□