What Joint Score Information Retains for Causal Ordering

Score-based methods can identify leaves in nonlinear additive noise models, but recovering a full causal order requires repeating this decision after each removal. Can the expected joint score-Jacobian matrix alone support this recursion? We show that it cannot in general, even when the matrix is known exactly. The proof uses two three-variable nonlinear models with equal-variance Gaussian noise. Their joint matrices are identical and identify X3 as a leaf in both models. After X3 is removed, one model requires X1 to precede X2, while the other requires X2 to precede X1. Since the matrix cannot distinguish these orders, no rule using it alone can guarantee a valid order over the model class. This limitation also applies to Schur elimination, although it is exact in linear Gaussian models. An exact identity shows that the Schur update differs from the true marginal matrix by a scaled covariance of the removed leaf's mechanism gradient. This term vanishes when the gradient is constant. When nonzero, it need not change the selected leaf, so experiments with Score-Schur Topological Sort (SSTS) test when the uncorrected update still yields valid orders. Across five paired synthetic datasets per dimension, SSTS gives zero edge violations at 50 variables but 4.6 +/- 2.4 at 100 variables, compared with 0.2 +/- 0.4 for public SCORE/DAS ordering. Separate paired diagnostics show that lower matrix-estimation error can accompany more order violations. A guarantee over the stated nonlinear model class still requires information beyond this joint expectation or further model restrictions.