Curvature-Controlled Transport Consensus Layer

1. Integration point: insert this as a residual dynamical layer after a graph message-passing block, or use it as the inner integrator of a manifold-valued optimizer. Start with sphere-valued embeddings, where exponential maps and parallel transport are available in closed form. Store each node state $x_i$ and tangent velocity $v_i$, compute geodesic distances, and define edge weights $a_{ij}$ from normalized attention or fixed graph adjacency. Use a barrier $B(r)=\mu r^2/(R^2-r^2)$ for $r<R$, with $R$ below the known injectivity radius. 2. Pseudocode: compute $d_{ij}$ and $P_{ij}v_j$; form $g_i=\sum_j a_{ij}(P_{ij}v_j-v_i)-\operatorname{grad}_{x_i}U$; update $v_i\leftarrow v_i+h g_i$; update $x_i\leftarrow\operatorname{Exp}_{x_i}(h v_i)$; transport the updated velocity to the new tangent space if required. Add the task loss after the dynamical block and backpropagate through a small number of integration steps. 3. Compute geodesics and transport exactly for the selected manifold; estimate $K_0$, $C_J$, and the injectivity margin from known geometry or local curvature probes. Log total energy $E$, maximum pairwise distance, and $D(t)=\max_{(i,j)}\|P_{ij}v_j-v_i\|$. 4. First cheap experiment: train a small sphere-valued graph neural network on Cora or synthetic clustered graphs, comparing Euclidean residual, chart-based Riemannian, and barrier-controlled transport layers. Sweep step size $h$ and coupling $a$. The predicted signature is a stability boundary when $C_J(K_0,R)Vh$ becomes order one, with $V$ the observed velocity bound; below it, energy should decrease up to discretization error and $D(t)$ should decay, while above it distance excursions and discrepancy spikes should occur. Test whether the measured boundary agrees with the estimated one within 20%.