BT-Critical Long-Memory Initialization

Integrate this at initialization of a small continuous-time RNN or structured state-space layer. Use a vector field such as $f(z)=Az+W_2\tanh(W_1z+c)$ and expose a scalar gain $\mu$ multiplying the designated two-dimensional center block. First choose parameters, solve for a reference equilibrium $z_*$, compute $J$, and calculate its characteristic invariants. Estimate a normalized null vector $q_0$ using the smallest right singular vector. Use JVPs through the Jacobian and invariant calculation to obtain $D_{q_0}\delta$ and $D_{q_0}\tau$. Optimize the initialization-only objective $L_{critical}$ for 100--500 steps while freezing task weights; reject updates for which $|e_{n-2}|<\epsilon$, then train normally while retaining $\mu$ as a learned or scheduled unfolding parameter. Compare vanilla initialization, eigenvalue-only near-critical initialization, and this BT initialization on a delayed-copy and damped-sinusoid task. Sweep $\mu$ and measure memory time, oscillation frequency, and rollout stability. The falsifiable prediction is that relaxation time increases as $|\delta|+|\tau|$ decreases, while the transition from monotone decay to oscillatory or growing trajectories occurs near the simultaneous invariant zero, not merely when the largest eigenvalue approaches zero. Fit local trajectories to the quadratic normal form and require recovered coefficients to agree with the invariant estimates within 20 percent.