Asymmetric Conservative Interaction Layer

Integrate this as a replacement for the pairwise message or update block in a particle simulator, velocity-grid neural operator, or mixture world model. Inputs are population distributions f_s[i] on a fixed velocity grid v_i, masses m_s, and timestep Delta t. First encode each population and use a small MLP to produce low-rank factors a_st,r[i] and b_st,r[j] for every ordered pair (s,t); do not tie the parameters or outputs for (s,t) to (t,s). Compute gradients with a fixed finite-difference matrix D. For each ordered pair, form G_st[i,j]=f_t[j](D f_s[i])/m_s-f_s[i](D' f_t[j])/m_t, contract over j with b_st,r[j], multiply by a_st,r[i], sum over r and t, and apply the divergence matrix D to obtain Q_s[i]. Set the provisional update delta_s=Delta t Q_s, flatten all species and grid cells into one vector, construct C from the mass, three momentum, and energy rows above, and replace the update by delta_star=delta-C^T(CC^T+epsilon I)^(-1)C delta. The paper supplies the divergence-form operator and ordered-kernel structure; the low-rank factorization, finite differences, and moment projection are implementation adaptations. Estimate entropy behavior empirically using H=sum_{s,i} f_s[i] log(f_s[i]+epsilon); reject or downweight steps where H increases, or add lambda max(0,H_{k+1}-H_k)^2 to training. Start with two species on a 1D or 2D velocity grid, a 32-channel factor MLP, and synthetic relaxation trajectories from a trusted conservative solver. Compare against an unconstrained pairwise MLP and a symmetric-kernel version at equal parameter count. Success means machine-precision conservation, fewer unstable rollouts, lower long-time distributional error, and comparable or lower wall-clock cost due to O(S^2 R N) contractions rather than O(S^2 N^2) pairwise interactions.