Lemma 9 ([The center-regular value equation]). There are a threshold j_ reg ≥j_ geom and constants 0<c_P<C_P such that the following holds for every j_0≥j_ reg , every sequence of parameters in K , and every 2≤l≤L . There is a unique …

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 therefore gives constants 0<λ_-I≤A_l(t)≤λ_+I. The radial condition b≥-1/4 , together with b≤0 , gives 0<p_-:=3m/4-1≤p(t)≤p_+:=m-1. Choose c_P>0 below all the initial values l so that λ_--p_+c_P-c_P^2>0 , and choose C_P above all of them so that λ_+-p_-C_P-C_P^2<0 . We may decrease c_P and increase C_P so that every θ_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^ T P_l'v ≥λ_--p_+c_P-c_P^2>0. At a contact with the upper barrier the corresponding derivative is at most λ_+-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 F_l(t) be the fundamental matrix of Y'=P_lY with F_l(2)=I . Its determinant is F_l(t) =(∫_2^ttrP_l(w)\,dw)>0. Moreover, Y''=(P_l'+P_l^2)Y=A_lY-pY', so its columns solve. On t≤2 they are e^ l(t-2) times constant vectors. Conversely, the two scalar radial factors for 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 F_l(t) gives the assertion about arbitrary value data. It remains to prove the reset estimate. The schedule gives t_j=(j^8)/(8)+O(j^7), _ t∈[t_j,t_ j+1 ] |(t)/(t_j)-1|⟶0. Times on period j and on the final rest immediately preceding it are therefore comparable to j^8 . From , a(t)-a_*=O(j^ -2 ), p(t)-p_∞=O(j^ -10 ) uniformly on these intervals. On a round interval set E_0=P_l-θ_l I . Subtracting from gives the exact difference equation E_0'=f_l^ rd (t)I-(pI+P_l)E_0-E_0θ_l I, f_l^ rd (t)=a(t)^ -2 B_l-p(t)θ_l-θ_l^2. Here f_l^ rd =O(j^ -2 ) by. The symmetric coefficients pI+P_l and θ_l I have a fixed positive sum of lower eigenvalue bounds. Lemma 8 ([A damped matrix difference]), on a final rest of length T_ j-1 , gives \|P_l(t_j)-θ_lI\| ≤C e^ -γT_ j-1 +Cj^ -2 =O(j^ -2 ). The discrepancy at the start of the rest is bounded by , 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 with the claimed uniformity.