Rank-One SRB Latent Dynamics

1. Integration point: implement this transition as the recurrent latent update in a world model, sequence autoencoder, or recurrent generator. Represent the angular state to the decoder as $(\cos\theta,\sin\theta)$, while retaining the scalar angle for derivative calculations. Use small spectrally normalized MLPs for $g_r$ and $g_y$. Parameterize $\rho=\operatorname{sigmoid}(a_\rho)$ and initially set $B=\tanh(a_B)I$, or project a learned $B$ after every update so that its spectral norm is below a chosen ceiling. 2. Pseudocode: initialize $(\theta,r,y)$; for each input $x_t$, calculate $\theta'=(\theta+\Omega+K\sin\theta+u_t)\bmod 2\pi$, $r'=\rho r+(1-\rho)g_r(\theta,x_t)$, and $y'=By+(1-\|B\|)g_y(\theta,r,x_t)$; decode $(\cos\theta',\sin\theta',r',y')$; add the task loss and optionally a weak penalty if the estimated angular exponent is outside a target interval. Estimate Lyapunov exponents by propagating Jacobian-vector products and applying QR reorthogonalization every few steps. 3. Computed directly from the mechanism: $\|B\|$, $\rho$, the angular derivative $1+K\cos\theta$, and finite-time exponent estimates. Estimated empirically: the invariant distribution, boundedness, mixing, and periodic windows. 4. First cheap experiment: compare this model with a GRU of equal parameter count on Lorenz-63 one-step prediction and 100-step rollout. Sweep $K\in[0,3]$ and $\rho\in\{0.8,0.9,0.98\}$, using the same optimizer and training budget. The quantitative prediction is a transition near the measured condition $\lambda_\theta=0$: above it, the largest exponent should become positive while transverse exponents remain negative; below it, trajectories should settle into periodic or quasiperiodic behavior. In periodic windows, $|(f^m)'(\theta^*)|<1$ should coincide with a negative largest exponent and a period-$m$ autocorrelation peak. Reject the mechanism if these transitions fail to align within approximately 20% with the exponent estimates or if latent norms are not bounded.