Dynamics-Matched Contractive Reservoir

1. Integration point: insert the module between an undersensed input encoder and the prediction head. Build a sparse graph reservoir with one latent state per entity or spatial cell. Each edge computes relative state features such as headway, velocity difference, and local density. Initialize the edge update with the IDM acceleration, then add a small trainable MLP residual. Use RK2, RK4, or explicit Euler integration. Start with a frozen reservoir and ridge-regression readout; only later unfreeze the residual vector field. 2. Pseudocode: initialize $x_0=0$; at each time step compute graph features, calculate the mechanistic acceleration and residual update, set $x_{k+1}=x_k+h f_\theta(x_k,u_k)$, append $z_k=[x_{k+1};u_k]$, and fit $W_{out}=YZ^T(ZZ^T+\lambda I)^{-1}$. During residual training, optimize the prediction loss plus the Jacobian penalty in the fourth formula. Estimate $\hat\rho$ with one or two power iterations using Jacobian-vector products, avoiding construction of the full Jacobian. 3. Computed from the mechanism: the IDM feature structure, interaction graph, contraction inequality, and closed-form linear readout. Estimated empirically: local Jacobian singular values, effective memory time, and actual alignment with the target dynamics. 4. First cheap experiment: use a 5-20 node traffic graph or synthetic coupled oscillators with 20% of states masked, comparing this model with a vanilla ESN, GRU, and graph neural ODE at equal latent width. Sweep Euler step size $h$ and residual scale. The falsifiable signature is a sharp instability boundary when $\hat\rho$ crosses one: hidden-state perturbations should decay geometrically with slope $\log\hat\rho$ below the boundary and grow above it. At $\hat\rho\approx0.9$, a perturbation should shrink by approximately $0.9^k$; measured decay rates should agree within 20%. Dynamics matching should reduce validation error and required readout samples relative to a random reservoir, especially at low observation density.