The Arithmetic of LLM Post-Training: Merging and Distillation on a Moving Recipient

Large language models are post-trained through successive stages such as SFT, RLVR, and OPD, yet it remains unclear how an update learned along one training path composes with a model that has already progressed along another. We study this question on Qwen3 models from 0.6B to 8B parameters by fixing an update learned in one training lineage (the donor) and carrying it, by merging or by distillation, to successive checkpoints of a second lineage (the recipient) that differs only in its random seed. The experiments show three regularities in how a donor update composes with its recipient. (1) Post-training updates are portable: a transferred update preserves the signature of its teacher, the problems it gains and the problems it loses. (2) Transfer is recipient-dependent: the same update improves an early recipient and harms a mature one, whether it was learned by reinforcement learning or by distillation. (3) Averaging and distillation diverge as the recipient matures: weight averaging leads from the stage where direct addition stops transferring, because it preserves more of the recipient's existing capability. A decomposition into problems that only the teacher solves and problems that only the recipient solves explains the second and third regularities: as the recipient advances, there is less to acquire and more to lose, and distillation keeps giving up the recipient's own problems while averaging retains them. These results show that the effect of a post-training update depends jointly on what the update encodes and what the recipient has already learned.