The gains of reinforcement learning with verifiable rewards (RLVR) can be reproduced by a very low-capacity steering-vector intervention whose effective control directions lie mainly in the low-variance complement of the activation principal subspace, with geometry that is stable within a task and base model

- Cai et al. freeze a model, add a single trainable vector per controlled layer, and distill an RL-tuned teacher into it: one vector per layer recovers over 85% of the base-to-RL performance gain on most evaluated tasks ("Effective Manifold Capacity"), with a lower bound on capacity below which intervention dimensionality and input-dependent expressiveness become constraints. [cai-etal-2026-geometry-of-rlvr] - "Control Manifold Separation": unconstrained steering directions put only about 1-2% of their energy in the top-10% principal subspace of the base activation covariance (below the 10% isotropic reference), the learned geometry is largely consistent across training configurations within a task and base model, and cross-task alignment correlates with capability transfer (six ordered topic pairs within the Science task, linear fit R^2 = 0.91). The 1-2% figure is measured for directions trained on four scientific topics; the five LLMs studied are DeepSeek-R1-Distill-Qwen-1.5B/7B and Qwen3-4B/8B/14B. These are observations about trained steering directions (the paper itself uses 'manifold' loosely for this low-dimensional control set), not proof of a natural geometry; the paper also proposes the Alpha-Stabler training controller. Limitation stated: only RLVR tasks (math, science, code, instruction following) were tested. [cai-etal-2026-geometry-of-rlvr]