Davis–Wielandt Shell Constraint for Heterogeneous SSMs

1. Integration point: use a heterogeneous recurrent or state-space model with modules indexed by r. Each module has local dynamics x⁽ʳ⁾_{t+1}=A_r x⁽ʳ⁾_t+B_r u⁽ʳ⁾_t and output y⁽ʳ⁾_t=C_r x⁽ʳ⁾_t+D_r u⁽ʳ⁾_t; couple outputs through u=-Ky+B_u v_t. Apply the constraint during optimizer steps or as a periodic projection of K. 2. Pseudocode: every M updates, freeze parameters and collect representative hidden states; linearize each nonlinear module around those states; evaluate G_r(e^{jω}) on a logarithmic grid using the displayed transfer formula; compute p_r=minω λ_min((G_r+G_r*)/2) and g_r=maxω σ_max(G_r); set p_blocks=min_r p_r and g_blocks=max_r g_r; calculate p_bound=p_blocks-||K||₂g_blocks². If p_bound<ε, scale K by α=min(1,(p_blocks-ε)/(||K||₂g_blocks²)) or reject the update. 3. The paper-derived quantities are shell coordinates, passivity margins, gain estimates, and the compositional inequality. The minibatch linearization, finite frequency grid, and frequency cutoff are empirical approximations. 4. First cheap experiment: compare a 2–4 block gated SSM or GRU on sequential MNIST and copy-memory against unconstrained coupling and spectral normalization. Sweep coupling scale α and record p_bound, hidden-state norm, Jacobian amplification, and long-horizon error. The quantitative prediction is a degradation boundary near p_bound=0: positive-margin runs should have bounded rollout amplification, while negative-margin runs should exhibit rapidly increasing hidden-state norms or prediction error. The measured critical coupling should be within 20% of ||K||₂=p_blocks/g_blocks².