Mapping-Cone Boundary Consistency Loss

Integrate this at the representation interface of a model with paired samples x in M and y = F(x) in N, such as a graph encoder with a node-to-graph map, a neural operator with interior and boundary coordinates, or a world model with a state-transition map. Add two small heads: a_theta(y), producing a vector field on target coordinates, and eta_theta(x), producing a scalar on source coordinates. For each minibatch, compute y = F(x), the Jacobian J_F(x) by autodiff or analytically, and the pullback residual r_boundary = J_F(x)^T a_theta(y) - grad_x eta_theta(x). For omega = a_i(y) d y_i, estimate the closure residual by autodiff: r_closed,ij = partial_{y_i} a_j - partial_{y_j} a_i. Optimize L = L_task + lambda_boundary times mean(||r_boundary|| squared) + lambda_closed times mean(sum_{i<j} r_closed,ij squared). Compact pseudocode is: y = F(x); a = a_theta(y); eta = eta_theta(x); J = jacobian(F, x); rb = transpose(J) @ a - grad(eta, x); rc = antisymmetric_jacobian(a, y); loss = task_loss + lb * mean(rb**2) + lc * mean(rc**2); backprop(loss). The paper supplies the exact mapping-cone residual; Jacobians, gradients, and residual norms are computed numerically. Since eta has gauge freedom up to an additive constant, center it per batch or impose zero mean. First test on a synthetic 2D-to-1D boundary map and on a small Poisson or advection neural-operator benchmark, comparing a standard supervised model against the same model with this loss. Measure test error, boundary or interface violation, robustness to noisy boundary labels, and gradient variance. A positive result is lower interface error and improved out-of-domain generalization at equal parameter count, without slowing the main task loss by more than 10 percent.