Transverse Synchrony Training

1. Integration point: use a continuous-time latent RNN, neural ODE, or discrete state-space model as the plant, with latent state $z_t$ and dynamics $z_{t+1}=F_\theta(z_t,u_t,v_t)$. Use a fixed or separately trained reference model $\zeta_{t+1}=F_{\mathrm{ref}}(\zeta_t,v_t)$. A controller produces $u_t=\pi_\phi(z_t,\zeta_t,v_t)$, and a synchrony residual is defined as $\epsilon_t=P(z_t)-P_\star(\zeta_t)$ for matched projections $P$ and $P_\star$. 2. Pseudocode: initialize $z_0$ and $\zeta_0$ independently; for each input $v_t$, compute $\zeta_{t+1}=F_{\mathrm{ref}}(\zeta_t,v_t)$, compute $\epsilon_t=P(z_t)-P_\star(\zeta_t)$, set $u_t=\pi_\phi(z_t,\zeta_t,v_t)$, and update $z_{t+1}=F_\theta(z_t,u_t,v_t)$. Optimize $L=L_{\mathrm{task}}+\alpha\sum_t\|\epsilon_t\|^2+\beta\sum_t\max(0,\widehat\lambda_{\perp,t}+\gamma)^2$, where $\gamma>0$ is a desired stability margin. 3. The paper supplies the invariant-manifold criterion $\epsilon=0$ and the requirement of asymptotic stability. Compute $\widehat\lambda_\perp$ empirically using automatic-differentiation Jacobian-vector products along the rollout, QR reorthogonalization, and perturbations restricted to the residual coordinates. 4. First cheap experiment: train a GRU or neural ODE on randomly forced Duffing or pendulum trajectories, comparing ordinary MSE with synchrony training. Initialize replicas near different equilibria and sweep coupling gain. The prediction is a sharp transition: when $\widehat\lambda_\perp>0$, residuals persist or diverge; when $\widehat\lambda_\perp<0$, residuals decay exponentially, with measured log-residual slope approximately equal to $\widehat\lambda_\perp$. Verify the predicted boundary within 20 percent by locating the gain where the exponent crosses zero.