Edge-Supported Polynomial State Space

1. Integration point: implement this as a fixed feature encoder and linear recurrent or ODE block inside a graph RNN or neural ODE. Given adjacency support $\mathcal E$, enumerate only base coordinates and active quadratic monomials such as $S_iI_j$; recursively construct degree-3 and higher monomials by differentiating the retained monomials under the known or learned quadratic interaction map. Use sparse COO indices for $T_d^{\mathcal E}$ and sparse block matrices for $K_d^{\mathcal E}$. 2. Pseudocode: initialize z=[x]; for k=2,...,d, append select_edge_monomials(z,E); compute dz=K_d@z; optionally compute r_hat=closure_mlp(z) only in the final block; integrate z <- solver_step(z,dz+r_hat,dt); decode the base state from the first $p$ coordinates. 3. Computed quantities are the graph support, sparse monomial indices, and exact linear/quadratic coefficient blocks when physical dynamics are known. For a learned model, estimate $K_d$ by least squares or gradient descent and estimate the closure from the observed residual $\dot z-K_dz$. 4. First cheap experiment: use a synthetic $n=32$ SEIR or quadratic graph system with an Erdős-Rényi or ring graph, comparing a dense polynomial lift, an edge lift, and a standard message-passing GNN at equal parameter count. Train one-step and rollout prediction. Measure dictionary size, wall-clock cost, rollout error, and $\|r_d\|_2$. The quantitative prediction is that degree-2 edge features scale linearly with $|\mathcal E|$ and remain far below the dense $p^2$ baseline; for fixed graph support, changing edge weights should not change feature dimension, while residual magnitude should increase with maximum weighted incoming intensity $\max_i\sum_j|a_{ij}|$ and approximately quadratically with state amplitude for the cubic omitted term.