Proof. The radial scale. Use the fixed smooth nondecreasing cutoff \(\chi\) from Section 3. For \(t>2\), put \[b(t)=-\mu\chi(t)(1+t)^{-5/4},\qquad a(t)=\exp\left(\int_2^t b(u)\,du\right),\] 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 (12) gives \(b\geq-1/4\). Since \(\chi'\geq0\), for \(t>2\) we have \[b'(t)\leq-\frac{5b(t)}{4(1+t)},\qquad b+b'+b^2 \leq b\left(1-\frac{5}{4(1+t)}+b\right)\leq0.\] The last factor is positive for \(t>2\) and \(b\geq-1/4\). The tail identities (23) 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 \[S=\frac12 g_r^{-1}\partial_r g_r =r^{-1}(I+B),\qquad B=\frac12\gamma^{-1}\dot\gamma=bI+H.\] Where a nonround part of a bump or a ramp occurs, \(J\) is fixed. Direct differentiation of \(G\) gives \[ H=\frac{\dot q}{2q}\left(P_J-\frac1mI\right),\qquad \operatorname{tr}H=0,\qquad \operatorname{tr}(H^2)=\frac{m-1}{4m}\left(\frac{\dot q}{q}\right)^2. \tag{24}\] 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 \[\operatorname{div}_\gamma(E\otimes E^\flat) =(\operatorname{div}_\gamma E)E^\flat +(\nabla^\gamma_EE)^\flat=0.\] Therefore \(\operatorname{div}_\gamma P_J=0\), and \(\operatorname{div}_\gamma H=0\), because the coefficients in (24) are constant on a slice. The trace \(\operatorname{tr}S=r^{-1}m(1+b)\) 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 \(\operatorname{Ric}_g(\partial_r,\partial_r)\). Tracing the tangential Gauss equation contributes \(S^2-(\operatorname{tr}S)S\) to the intrinsic slice Ricci endomorphism. Adding the normal contribution gives \[\operatorname{Ric}_g^\#\big|_{TS^m} =\operatorname{Ric}_{g_r}^\#-\partial_rS-(\operatorname{tr}S)S.\] Here an index of the tangential tensor is raised using \(g_r\). The mixed formula is \(\operatorname{div}_{g_r}S-d(\operatorname{tr}S)\), already shown to vanish. Substitution of \(S=r^{-1}(I+B)\), and \(\operatorname{tr}B=mb\), yields \[ \begin{aligned} r^2\operatorname{Ric}_g(\partial_r,\partial_r) &=-m(b+\dot b+b^2)-\operatorname{tr}(H^2),\\ r^2\operatorname{Ric}_g^\#\big|_{TS^m} &=\operatorname{Ric}_\gamma^\#-\dot B -\bigl(m(1+b)-1\bigr)(I+B). \end{aligned} \tag{25}\] 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, \[|q-1|=O(T_j^{-1}),\qquad |\dot q|=O(T_j^{-2}),\qquad |\ddot q|=O(T_j^{-3}).\] 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 (24) gives, throughout period \(j\), \[\|H\|=O(T_j^{-1}),\qquad \|\dot H\|=O(T_j^{-2}),\qquad \operatorname{tr}(H^2)=O(T_j^{-2})=O(j^{-12}).\] The stronger pulse bound is \(\operatorname{tr}(H^2)=O(T_j^{-4})\). 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 (22), \[t_j\sim\frac{j^8}{8},\qquad \sup_{t\in[t_j,t_{j+1}]}\left|\frac{t}{t_j}-1\right|\longrightarrow0.\] On the scale tail, an exact calculation gives \[-m(b+\dot b+b^2) =m\mu(1+t)^{-5/4} \left(1-\frac{5}{4(1+t)}-\mu(1+t)^{-5/4}\right).\] 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 \(\operatorname{tr}(H^2)\). The radial component in (25) is therefore positive once \(j_0\) is large enough.

For the tangential component, the strict inequality (3) on the compact interval \([1,q_*]\) implies that, for some fixed \(\delta>0\), \[\operatorname{Ric}_{G(a_*,q,J_0)}^\#\geq(m-1+2\delta)I \qquad(1\leq q\leq q_*).\] 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 \[\operatorname{Ric}_\gamma^\#\geq(m-1+\delta)I.\] On the other hand, \(B=bI+H\), \(b,\dot b\to0\), and the displayed bounds for \(H,\dot H\) show uniformly that \[\dot B+\bigl(m(1+b)-1\bigr)(I+B) =(m-1)I+o(1).\] The tangential component of (25) 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 \(\operatorname{Ric}_g\geq0\) 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: \[g=dr^2+f(r)^2g_0,\qquad f(r)=r\,a(\log r).\] The scale construction gives \[f'(r)=a(1+b)\in(0,1],\qquad f''(r)=\frac{a}{r}(b+\dot b+b^2)\leq0.\] 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.

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 \(\mathbb R^{m+1}\), so the metric is complete. The underlying manifold is connected, smooth, and without boundary. ◻

Proof. The Frobenius inner product gives \[\frac12\frac{d}{dt}\|E\|^2 =\langle E,F\rangle-\langle E,\mathsf L E\rangle -\langle E,E\mathsf R\rangle \leq \|E\|\,\|F\|-\gamma\|E\|^2.\] 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 (30). Integrating that estimate in \(t\) and interchanging the nonnegative integrals gives (31). ◻

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 (4) therefore gives constants \[0<\lambda_-I\leq A_l(t)\leq\lambda_+I.\] The radial condition \(b\geq-1/4\), together with \(b\leq0\), gives \[0<p_-:=3m/4-1\leq p(t)\leq p_+:=m-1.\] 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 \[v^{\mathsf T}P_l'v \geq\lambda_--p_+c_P-c_P^2>0.\] 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 \[\det\mathcal F_l(t) =\exp\left(\int_2^t\operatorname{tr}P_l(w)\,dw\right)>0.\] Moreover, \[Y''=(P_l'+P_l^2)Y=A_lY-pY',\] so its columns solve (32). On \(t\leq2\) they are \(e^{l(t-2)}\) times constant vectors. Conversely, the two scalar radial factors for (32) 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 \[ t_j=\frac{j^8}{8}+O(j^7),\qquad \sup_{t\in[t_j,t_{j+1}]}\left|\frac{t}{t_j}-1\right|\longrightarrow0. \tag{36}\] Times on period \(j\) and on the final rest immediately preceding it are therefore comparable to \(j^8\). From (23), \[ a(t)-a_*=O(j^{-2}),\qquad p(t)-p_\infty=O(j^{-10}) \tag{37}\] uniformly on these intervals. On a round interval set \(E_0=P_l-\theta_l I\). Subtracting (26) from (33) gives the exact difference equation \[E_0'=f_l^{\mathrm{rd}}(t)I-(pI+P_l)E_0-E_0\theta_l I,\qquad f_l^{\mathrm{rd}}(t)=a(t)^{-2}B_l-p(t)\theta_l-\theta_l^2.\] Here \(f_l^{\mathrm{rd}}=O(j^{-2})\) by (37). The symmetric coefficients \(pI+P_l\) and \(\theta_l I\) have a fixed positive sum of lower eigenvalue bounds. Lemma 11, on a final rest of length \(T_{j-1}\), gives \[\|P_l(t_j)-\theta_lI\| \leq C e^{-\gamma T_{j-1}}+Cj^{-2}=O(j^{-2}).\] The discrepancy at the start of the rest is bounded by (34), 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 (35) with the claimed uniformity. ◻

Proof. The scalar barriers used in Lemma 12 also keep \(v_l\) between fixed positive constants. Subtracting (26) from (38) gives \[(v_l-\theta_l)'= \bigl(a(t)^{-2}B_l-p(t)\theta_l-\theta_l^2\bigr) -(p+v_l+\theta_l)(v_l-\theta_l).\] Its initial value is zero. The forcing is \(O(j^{-2})\) by (37), and the damping coefficient is uniformly positive. Thus \[ v_l-\theta_l=O(j^{-2}) \quad\text{throughout the pulse}. \tag{40}\] We retain \(v_l\) exactly in the value equation: even the small pointwise difference in (40) can have a large integral over \(T_j\).

For the angular operator, expand (4) at \(q=1\). Its first derivative in \(q\) is \[a^{-2}\left(\frac{B_l}{m}I+D_J^2|_{V_l}\right).\] In particular the trace-free part has the positive coefficient \(a^{-2}(D_J^2)_{\mathrm{tf}}\) that appears in (20). Since \(q=1+z/T\) on the pulse and \(a(t)-a_*=O(j^{-2})\), Taylor’s formula gives \[ \begin{aligned} A_l(t,h) &=a(t)^{-2}B_lI+\frac{p_\infty+2\theta_l}{T}R_l(\tau,h) +\mathcal E_{A,l}(t,h),\\ \|\mathcal E_{A,l}\|_{C^1_h} &\leq C\left(\frac{j^{-2}}{T}+\frac1{T^2}\right). \end{aligned} \tag{41}\] 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 \[Q_l(t,h)=v_l(t)I+\frac1T R_l(\tau,h).\] It is symmetric, and it has a fixed positive lower eigenvalue bound when \(j\) is large. Its Riccati residual is \[\rho_l=A_l-pQ_l-Q_l^2-Q_l'.\] The scalar terms in this expression cancel by (38). Using \(\partial_t(R_l/T)=\partial_\tau R_l/T^2\), the remaining identity is \[\rho_l =-\frac{p-p_\infty+2(v_l-\theta_l)}{T}R_l -\frac{R_l^2+\partial_\tau R_l}{T^2} +\mathcal E_{A,l}.\] Equations (40) and (41) therefore imply \[ \sup_{t,h} \bigl(\|\rho_l\|+\|D_h\rho_l\|\bigr) \leq\frac{C\eta_j}{T}. \tag{42}\]

Let \(E_l=P_l-Q_l\). No matrices are commuted in the exact difference equation \[ E_l'=\rho_l-(pI+P_l)E_l-E_lQ_l. \tag{43}\] The lower eigenvalue bounds for \(P_l,Q_l\), and \(p\) allow Lemma 11 with a fixed damping rate. The profiles vanish near \(\tau=0\), so \[E_l(t_j)=P_l(t_j)-\theta_lI=O(j^{-2})\] by (35). From (30), (31), and (42), we obtain \[ \sup_{t,h}\|E_l(t,h)\|\leq C\eta_j,\qquad \sup_h\int_{t_j}^{t_j+T}\|E_l(t,h)\|\,dt\leq C\eta_j. \tag{44}\] 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 (43) and using \(D_hP_l=E_{l,h}+Q_{l,h}\) gives the exact regrouping \[ E_{l,h}'=D_h\rho_l -(pI+P_l)E_{l,h}-E_{l,h}P_l -Q_{l,h}E_l-E_lQ_{l,h}. \tag{45}\] 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 11 to (45). Its forcing has pointwise norm at most \(C\eta_j/T\), by (42) and the first bound in (44). Its integrated norm is at most \[C\eta_j+\frac{C}{T} \int_{t_j}^{t_j+T}\|E_l\|\,dt \leq C\eta_j.\] Consequently \[ \sup_{t,h}\|E_{l,h}\|\leq\frac{C\eta_j}{T},\qquad \sup_h\int_{t_j}^{t_j+T}\|E_{l,h}\|\,dt\leq C\eta_j. \tag{46}\] 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 \[F_l(t,h)= e^{-\int_{t_j}^t v_l(w)\,dw}\mathcal B_l(t,h),\qquad G_l(t,h)=H_l((t-t_j)/T,h).\] Then \[F_l'=(R_l/T+E_l)F_l,\qquad G_l'=(R_l/T)G_l,\qquad F_l(t_j)=G_l(t_j)=I.\] 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 \[Z_l'=(R_l/T+E_l)Z_l+E_lG_l,\qquad Z_l(t_j)=0.\] The integral bound for \(E_l\) gives \(\sup_t\|Z_l\|\leq C\eta_j\). Differentiating this equation introduces the additional terms \[(D_hR_l/T+E_{l,h})Z_l+E_{l,h}G_l+E_lD_hG_l.\] Their integrated norms are \(O(\eta_j)\), using the preceding uniform bounds and (46). The same integral inequality gives \(\sup_t\|D_hZ_l\|\leq C\eta_j\). Evaluation at the pulse end proves the first assertion of (39). At that end \(R_l(1,h)=D_hR_l(1,h)=0\). Combining \(P_l=Q_l+E_l\), (40), (44), and (46) proves the second assertion. Every estimate used the earlier history only through the uniform reset and barriers of Lemma 12, 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 \[y''+p(t)y'-A_i(t)y=0,\qquad A_i(t)>0.\] Multiplying the equation for \(y'\) by the positive integrating factor \(e^{\int p}\) gives \[\bigl(e^{\int p}y'\bigr)'=e^{\int p}A_i y.\] 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 (47) gives positivity of \(D_{l,j}\), and entry by entry \[0<\frac{(V_{l,j})_{ii}} {(U_{l,j})_{ii}+\theta_l(V_{l,j})_{ii}} \leq\theta_l^{-1}.\] This proves (48).

For an incoming value vector \(y\) at the start of the period, the data at the pulse end are \[\bigl(\mathcal B_{l,j}y,\, P_{l,e}\mathcal B_{l,j}y\bigr).\] Their value at the end of the leg is \[\mathcal T_{l,j}y =\bigl(U_{l,j}+V_{l,j}P_{l,e}\bigr)\mathcal B_{l,j}y.\] Since \(D_{l,j}=U_{l,j}+\theta_lV_{l,j}\), left multiplication by \(D_{l,j}^{-1}\) gives (49). 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 (48).

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 \[w'=A_{l,i}(t)-p(t)w-w^2.\] The scalar barrier argument in Lemma 12, with initial value \(\theta_l\), keeps \(w\) between fixed positive constants. During the dwell, \[A_{l,i}(t)=\lambda_{l,i}+O(j^{-2}),\qquad \lambda_{l,i}=d_{l,i}(d_{l,i}+p_\infty),\] by (37) and the fixed value \(q=q_*\). For \(e=w-d_{l,i}\), subtraction gives \[e'=F_{l,i}(t)-(p+w+d_{l,i})e,\qquad F_{l,i}=A_{l,i}-\lambda_{l,i} -(p-p_\infty)d_{l,i}=O(j^{-2}).\] The value of \(e\) at the dwell entrance is bounded, and its damping coefficient has a fixed positive lower bound. The scalar version of (31) yields \[\int_{\text{dwell }j}|w-d_{l,i}|\,dt \leq C+C L_jj^{-2}.\] 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 (50). 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 (49), \[e^{-\varphi_{l,j}}D_{l,j}^{-1}\mathcal T_{l,j}(h) =\left[I+K_{l,j}\bigl(P_{l,e}(h)-\theta_lI\bigr)\right] e^{-\varphi_{l,j}}\mathcal B_{l,j}(h).\] The bound (48) and Lemma 13 imply \[ e^{-\varphi_{l,j}}D_{l,j}^{-1}\mathcal T_{l,j}(h) =H_l(1,h)+O_{C^1}(\eta_j). \tag{52}\] 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 (52) gives \[ \Psi_j(h):= \bigl(\mathcal N_l(D_{l,j}^{-1}\mathcal T_{l,j}(h))\bigr)_{l=2}^{L} =\Phi(h)+O_{C^1}(\eta_j) \quad\text{in }\mathcal G. \tag{53}\] 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 (29), \(A_0\) 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 \[\sup_{h\in\overline B_r} \|I-A_0^{-1}Df(h)\|\leq\frac14.\] For every sufficiently large \(j\), uniformly in the admitted past, \(f_j=\zeta\circ\Psi_j\) is defined on this ball and \[\sup_{h\in\overline B_r} \|A_0^{-1}(Df_j(h)-Df(h))\|\leq\frac14,\qquad \|A_0^{-1}f_j(0)\|\leq\frac r2.\] This follows from (53) and \(\eta_j\to0\). The map \[\mathcal C_j(h)=h-A_0^{-1}f_j(h)\] has derivative of norm at most \(1/2\) on the ball. Also \[\|\mathcal C_j(h)\| \leq\|\mathcal C_j(0)\|+\tfrac12\|h\| \leq r \qquad(h\in\overline B_r).\] 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 10. Undoing determinant normalization at each fixed point gives \[D_{l,j}^{-1}\mathcal T_{l,j}=s_{l,j}\Pi_l,\qquad s_{l,j}=\det(D_{l,j}^{-1}\mathcal T_{l,j})^{1/N_l}>0.\] By (52), 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 \[\log s_{l,j}=\varphi_{l,j}+O(1)=O(T_j),\] because \(v_l\) is uniformly bounded. This proves (51). The equality is for the complete period; no residual change of direction is passed to a later period. ◻

Proof of Theorem 1. We convert the exact cycles from Proposition 15 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 12 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 (32), 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 (51) 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 \[\varepsilon_{l,i,j} =L_j^{-1}\log(D_{l,j})_{ii}-d_{l,i}, \qquad \max_{l,i}|\varepsilon_{l,i,j}|\longrightarrow0\] by (50), one has \[ \log\|Y(t_J)\|= \sum_{j=j_0}^{J-1} \left(\log s_{l,j} +j^7\bigl(d_{l,i_{j+1}}+\varepsilon_{l,i_{j+1},j}\bigr)\right). \tag{54}\] Signs in \(\Pi_l\) do not affect this norm.

We verify the weighted average in (54). 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 \[\sum_{j=j_0}^{J-1}j^7d_{l,i_{j+1}} =\overline d_l\sum_{j=j_0}^{J-1}j^7+O(J^7).\] 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, \[\sum_{j=j_0}^{J-1}j^7=\frac{J^8}{8}+O(J^7),\qquad t_J=\frac{J^8}{8}+O(J^7)\] by (22). Therefore \[ \lim_{J\to\infty}\frac{\log\|Y(t_J)\|}{t_J} =\overline d_l<k. \tag{55}\]

The estimate extends to every radius. From \(Y'=P_lY\) and (34), \[\|Y(t)\|\leq \exp\bigl(C_P(t-t_j)\bigr)\|Y(t_j)\| \qquad(t_j\leq t\leq t_{j+1}).\] The ratio \((t_{j+1}-t_j)/t_j\) tends to zero by (36). Thus (55) gives \[\limsup_{t\to\infty}\frac{\log\|Y(t)\|}{t} \leq\overline d_l.\] 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 \[\left|\sum_iY_i e_{l,i}(\omega)\right|\leq C_l\|Y\| \quad\text{for every }\omega\in S^m,\] 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 \[\dim_{\mathbb R}\mathcal H_k(\mathbb R^{m+1},g) \geq\sum_{l=2}^{L}M_l>\sum_{l=0}^{k}N_l.\] 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 10 give \[\operatorname{vol}_g B_g(0,R) =\operatorname{vol}_{g_0}(S^m) \int_0^R r^m a(\log r)^m\,dr.\] 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 \[W_{0a0b} =-\frac{1}{m-1} \left(\operatorname{Ric}_{h_*}-\frac{\operatorname{Scal}_{h_*}}{m}h_*\right)_{ab}.\] 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 \[\alpha_l(q)(\alpha_l(q)+m-1) =a_*^{-2}q^{1/m} \bigl[l(l+m-1)+(q^{-1}-1)l^2\bigr].\] 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 \[\alpha_l(1)=a_*^{-1}l+O(1),\qquad \alpha_l(q_*)=a_*^{-1}q_*^{-(m-1)/(2m)}l+O(1).\] 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, (13) 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] 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 (55), need not equal the local homogeneous degree seen after normalization on each separate cone subsequence. ◻