Mass-Conserving Puncta Router

(1) Integration point: place the module after a token, image, or graph encoder and before an MoE router or attention block. Let h_i\in\mathbb R^d be the feature at node i, and let the module output p_i=C_i/(C_i+C_i^{\dagger}+\epsilon), q_i=C_i+C_i^{\dagger}, and optionally q_i h_i for routing. Use a 1D sequence Laplacian, 2D four-neighbor image Laplacian, or normalized graph Laplacian. Initialize A_i,B_i,A_i^{\dagger},B_i^{\dagger} with softplus projections of h_i plus a positive floor, and initialize C_i=C_i^{\dagger}=0. (2) Pseudocode: compute k_i=k_0\exp(g_k(h_i)) and v_i=v_0\exp(g_v(h_i)); for t=1,...,S compute rC_i=k_i A_i B_i^{\dagger}-v_i C_i and rCd_i=k_i A_i^{\dagger}B_i-v_i C_i^{\dagger}; update each state with z_i\leftarrow z_i+dt[D_z(Lz)_i+r_{z,i}], using r_A=-rC, r_{B^{\dagger}}=-rC, r_{A^{\dagger}}=-rCd, and r_B=-rCd; apply a positivity-preserving clamp or softplus parameterization after each step. For improved conservation, use reaction splitting: perform the local reaction update in several small substeps, then apply conservative graph diffusion. (3) The paper-derived quantities are the reaction stoichiometry, zero-row-sum diffusion, and conserved mass identities. Estimate k_0,v_0,D_A,D_B,D_C and the rate networks by backpropagation. Enforce an explicit-step stability condition dt\max_z D_z\rho(-L)<1 and dt\max_i(k_i\max(A_i,B_i)+v_i)<1, or use implicit diffusion. Monitor M_{A+C}=\sum_i(A_i+C_i), M_{B^{\dagger}+C}=\sum_i(B_i^{\dagger}+C_i), M_{A^{\dagger}+C^{\dagger}}, and M_{B+C^{\dagger}}; relative drift should remain below 10^{-5}. (4) First cheap experiment: use CIFAR-10 or MNIST patches with a small CNN or ViT, comparing a standard softmax top-1 router and an unconstrained sigmoid router against this layer at equal parameter count and FLOPs. Measure classification accuracy, assignment entropy, connected-component count, orientation overlap \sum_i p_i(1-p_i), and robustness to feature noise. A successful result is lower overlap between opposite orientations, more spatially coherent active regions, and equal-or-better accuracy at the same number of routed experts. For language, repeat on a synthetic sequence task where contiguous spans share a latent label; success is higher span consistency without training-loss oscillation.