Flat Task-Transport Connection

Integrate the method at the fine-tuning or conditional-hypernetwork level. Let z in R^d be a known task or domain embedding, let w be a shared model's LoRA adapter weights or another small trainable parameter vector, and let a transport module output d vector fields A_i(w,z). At each training step, sample z and two distinct coordinate directions i,j. Compute the task loss using the current adapter weights. Use automatic differentiation to obtain partial_{z_i}A_j and partial_{z_j}A_i, and compute the two Jacobian-vector products D_wA_j[A_i] and D_wA_i[A_j] without forming a dense Jacobian. Construct F_ij using the displayed curvature formula and add lambda_flat times its squared norm to the task loss. Also sample a small epsilon and evaluate a finite holonomy diagnostic: transport first in direction i and then j, transport first in direction j and then i, and measure the norm of the difference. Pseudocode: sample z,i,j; calculate A_i and A_j; calculate the four derivative or Jacobian-vector terms; set flat_loss = norm(partial_i_Aj - partial_j_Ai + Jw_Aj(A_i) - Jw_Ai(A_j))^2; update the transport module and adapter using task_loss + lambda_flat*flat_loss. The paper's mathematical input is the splitting and Lie-bracket curvature condition; all derivatives are estimated by autodiff. Begin with a two-dimensional rotated-MNIST or permuted-MNIST task manifold, a small MLP, and LoRA adapters. Compare independent adapters and an unconstrained hypernetwork at matched parameter count. Success means lower direct-versus-two-step transport discrepancy, lower accuracy variance under different task traversal orders, and better accuracy on held-out intermediate task coordinates at equal training compute.