What Will Post-Training Fix? Per-Problem Gains Are Shared Across Independent RL Runs, and Existing Checkpoints Predict Them Better Than A Priori Signals
Data selection, curricula and the evaluation of post-training recipes all assume that we can tell, before training, which problems a model will improve on. We test this assumption directly. For two base models, DeepSeek-R1-Distill-Qwen-1.5B and Qwen2.5-Math-1.5B, we evaluate eighteen post-training runs on up to 1532 competition math problems with many samples per problem, and compare signals available before training against a noise ceiling derived from the agreement between disjoint subsets of runs. Three findings hold on both base models. What post-training fixes is shared: independent runs agree on which rarely solved problems improve, with a noise ceiling of about 0.9, yet the two base models agree with each other only at rho=0.25 - the shared component belongs to the base model, not to the problem. A priori signals capture a minority of it: base pass rate, the likelihood of a correct solution, a larger model's pass rate and their combinations explain only 0.30 and 0.24 of the explainable variance in gains. Existing checkpoints are the better predictor: the per-problem gains of a single checkpoint from another family predict a new run better than every a priori signal, alone or combined (0.52 vs. 0.33 and 0.34 vs. 0.17 against the combined signals). The conclusions hold on problems from 2025-2026 competitions and when the baselines on the two sides of every comparison are estimated independently. We propose the noise ceiling as a standard companion to per-problem signals.