The Geometry of Sequential Learning: Lie-Bracket Prediction of Transfer Order

Sequential learning is order-dependent: from Pile-style next-token domain adaptation to instruction-SFT and DPO, $N$ candidate sources induce $N!$ possible curricula. We show that the local order effect is governed by a computable geometric quantity, the Lie-bracket commutator of gradient update fields, yielding a pairwise score for whether $A \to B$ or $B \to A$ is better for a target domain. The pairwise bracket primitive also defines a Lie-Bracket Tournament: with a shared $\theta_0$ target-gradient reference, Hessian symmetry gives Borda/row-sum scores from one Hessian-vector product per source, $O(N)$ dot products, and an $O(N\log N)$ sort, without materializing the $O(N^2)$ edge matrix. Empirically, the planner reaches 98.1%/98.9% pairwise accuracy at $k=1$ for instruction-SFT/DPO, remains at 73.1%/72.2% at $k=20$, and preserves the original pretraining-domain evidence with 82.4–92.0% accuracy across four LLMs and 91.1% on diffusion. At curriculum scale, it recovers the best of all $3!$ schedules in 87.5% of trials, ranks 85 Stack programming-language source domains for a Python target in the 99th sampled percentile, and reaches the 99.0–99.6th sampled percentile on 56 MMLU subjects, sharply above the reported descending gradient-norm baseline. These results reframe sequential learning as a geometric tournament problem: commutators provide both local pairwise order information and a scalable primitive for many-domain schedules.