Tangential Landau Pairwise Noise

Integrate this at the stochastic interaction or transition stage of a particle graph neural network, particle world model, or diffusion sampler. Let each particle carry position $q_i$, velocity $v_i$, and learned hidden state $h_i$. The neural network may still produce a deterministic pair drift, but the proposed stochastic term is added only to the velocity channel. For each sampled edge $(i,j)$, compute $r=v_i-v_j$, $A_\varepsilon=(\|r\|^2+\varepsilon^2)^{(2+\gamma)/2}$, and $P_\varepsilon=I-rr^\top/(\|r\|^2+\varepsilon^2)$. Draw $z\sim\mathcal N(0,I_d)$ and set $u=\sqrt{2\eta A_\varepsilon}P_\varepsilon z$. If the learned pair drift is $g_{ij}=\mathrm{MLP}([h_i,h_j,r,q_i-q_j])$, apply $v_i\leftarrow v_i+\eta g_{ij}+u$ and $v_j\leftarrow v_j-\eta g_{ij}-u$ using the same noise sample for both endpoints. For exact energy preservation of the noise-only step, define $c=(v_i+v_j)/2$, $r'=r+2u$, and replace $r'$ by $r''=\|r\|r'/\max(\|r'\|,\delta)$ before setting $v_i=c+r''/2$ and $v_j=c-r''/2$. The paper supplies the tangential covariance and soft-potential scaling; $\varepsilon$, $\eta$, edge sampling, and energy correction are tunable. The first experiment should use a 2D or 3D spring-particle or charged-particle rollout task with a small equivariant graph baseline. Compare isotropic Gaussian noise, unconstrained learned pair noise, and the Landau module at equal parameters and FLOPs. Measure center-of-mass drift, kinetic-energy error, long-horizon rollout error, and predictive negative log-likelihood. Success means materially lower invariant drift and improved long-horizon stability without degrading one-step prediction; a target is at least 2x lower center-of-mass or energy error at equal rollout accuracy.