Compositional Generalization Certificates via the Van Kampen Theorem

A neural network trained on the multiplication table of the quaternion group Q₈ can achieve perfect accuracy within each cyclic subgroup ⟨i⟩ and ⟨j⟩, predict every cross-region product wrong, and stay there indefinitely. This is the grokking plateau, and we prove its height is exactly 4 in dimension d=2, determined by the topology of the overlap before any training. The mechanism is a homotopy pullback: for G = A ∗_C B with finite overlap, the representation space of G is a homotopy pullback of those of A and B over that of C, so locally accurate modules need only agree up to a basis change. From this we derive a stratified generalization certificate whose strata are the connected components of Hom(C, U(d)): in the correct stratum, with locally accurate modules, the cross-region prediction is O(ε)-close to the target; in the wrong stratum it is provably far regardless of local accuracy, with exact gap c = 2(d − rₘₐₓ) for central cyclic overlap. The plateau is therefore the time the network spends in the wrong stratum, the grokking transition is the discrete jump between strata, and the framework predicts which task families exhibit such plateaus—those whose overlap representation space is disconnected—and which do not.