Conservative amortized collision layer

Integrate the layer at the interaction stage of a particle graph neural network. Each particle stores position $x_i$, velocity $v_i$, and feature $h_i$. Sample or construct candidate pairs $(i,j)$ and compute $h_{ij}=\mathrm{MLP}([h_i,h_j,x_i-x_j,\|x_i-x_j\|,\|v_i-v_j\|])$. For each pair, calculate $V=(v_i+v_j)/2$, $r=v_i-v_j$, and $u=r/(\|r\|+\epsilon)$. A conditional flow receives $(u,h_{ij})$ and samples $z\sim\mathcal N(0,I_{d-1})$. Map $z$ through $T_\phi$ to an unconstrained vector, normalize it to obtain $u'$, and set $r'=\|r\|u'$. Apply the deterministic reconstruction $v_i\leftarrow V+r'/2$ and $v_j\leftarrow V-r'/2$. If interactions should be probabilistic, predict $p_{ij}=\sigma(g_\phi(h_{ij}))$ and use an identity update with probability $1-p_{ij}$; otherwise always apply the transition. Do not add an unconstrained velocity residual after reconstruction. In three dimensions, represent $u'$ with a local two-dimensional tangent-plane chart or a normalized three-vector; handle near-zero relative velocity by skipping the pair. Train the flow with conditional negative log likelihood on observed pre/post pairs, optionally adding downstream rollout loss. The mathematical guarantees are the center-of-mass decomposition and exact invariant-preserving reconstruction. The Green's function is approximated by the flow, while pair rates, pair sampling, chart parameterization, and network capacity are empirical. First test on a 32- or 64-particle toy system with known elastic collisions. Compare against an unconstrained interaction GNN, a Gaussian residual transition, and a hand-coded elastic-collision solver at equal parameter count and similar pair-update cost. Record one-step transition NLL, long-horizon trajectory error, total momentum drift, and kinetic-energy drift. The first success criterion is machine-precision conservation; stronger evidence is lower transition NLL and lower long-horizon error than the unconstrained and Gaussian baselines at equal compute.