What Determines the Direction of Post-Training Gains? A Preregistered Multi-Scale Study of Method, Scale, and Base-Model Identity

Preprint of the findings paper. Companion methods paper: "Benchmarking Post-Training Under Constrained Compute: Protocols for Question Generation, Variance-Controlled Evaluation, and Judge Auditing" (Zenodo, DOI: 10.5281/zenodo.23409480). In an earlier pilot on a smaller single-judge pool, fine-tuning our own 4B model sharply lowered its perplexity on the target corpus without a corresponding gain in its judged score, motivating the question of what determines the direction and magnitude of the post-training gain. We answer it with a two-sided design: a within-family elasticity surface over four scale points crossed with three training methods, and two generalization probes, a second base family and a second task domain. The surface reverses the ordering of the two reinforcement-style methods between 4B and 9B (RLVR-C v2 ahead at 4B, MGPO-EG ahead at 9B) and yields one preregistered adjacent-granularity promotion, reported as a near-threshold result rather than a significance claim. Both probes return a gain-sign flip that tracks base-model identity, and an ablation demonstrates that the active ingredient of our momentum-perturbed optimizer is the perturbation itself.