C1 Homogeneous Lyapunov Critic

1. Integration point: use this as a Lyapunov critic for a neural ODE dot z=f_theta(z), an SSM transition, or the continuous interpolation of an RNN update. Restrict the critic to degree two, V_phi(z)=||z||^2 q_phi(u), with u=z/(||z||+epsilon). Represent q_phi on the circle or sphere using periodic cubic B-splines, Fourier features followed by softplus, or a small angular MLP; enforce q_phi(u)>=q_min>0 with q_phi=softplus(h_phi)+q_min. 2. Pseudocode: sample states z_i from training trajectories and random perturbations; compute u_i=z_i/(||z_i||+epsilon), V_i=||z_i||^2 q_phi(u_i), and D_i=gradient_z V_phi(z_i)^T f_theta(z_i) by automatic differentiation. Minimize task loss plus lambda times softplus(D_i+alpha V_i^((d+1)/2)), together with penalties for q_phi(u_i)<q_min and angular derivative discontinuities. For the cubic case set d=3, so the target is D_i<=-alpha V_i^2. 3. Computed from the paper: homogeneity, the Lie-derivative certificate, and the algebraic decay exponent. Estimated empirically: alpha, the minimum sphere margin, and C1 behavior near the origin. 4. First cheap experiment: train a two-dimensional neural ODE on a stable synthetic trajectory-reconstruction task, comparing a polynomial quadratic or quartic Lyapunov critic with the angular C1 critic. Roll out from radii 10^-2 through 10^2. The quantitative prediction is that successful cubic dynamics satisfy a log-log slope d log V/d log t approximately -1 for the quadratic certificate, while polynomial-certificate training can fail despite bounded trajectories; the measured slope should approach -1 over at least one decade and the estimated sphere margin -D/V^2 should remain positive.