Beyond Pairwise Task-Order Models: A Seed-Aware Spectral Audit of Continual-Learning Landscapes

Task order can change catastrophic forgetting, yet a good position-effect or transition model does not establish that pairwise structure is exactly sufficient. We introduce a seed-aware spectral audit for complete task-order experiments. The audit treats a scalar response over all task permutations as a function on the symmetric group, decomposes it by assignment degree, removes the diagonal seed-variation term when estimating reproducible signal, and jointly tests the entire higher-degree complement. In ten-seed MLP experiments containing every order, degree greater than two accounts for $16.1\%$ (approximate $95\%$ CI $[12.1\%,20.1\%]$) of estimated nonconstant signal at $K=5$ and $19.4\%$ ($[11.4\%,27.4\%]$) at $K=6$. Conditional on independent seed-level central symmetry, both whole-seed randomization tests reach the ten-seed resolution limit, $p=1/512$. A separate, explicitly post-selected analysis uses a complete $K=7$ landscape to illustrate how the audit localizes the residual within degree three. The result is a reusable distinction between low-degree approximation strength and exact structural sufficiency in continual-learning order effects.