Sharpen Before You Adapt: Data-Free Entry-State Sharpening for Test-Time Reinforcement Learning

Test-time reinforcement learning (TTRL) adapts language models on unlabeled test problems using supervision derived from their own samples. This makes the checkpoint's \emph{entry state} consequential: a diffuse policy provides noisier self-supervision and may spend much of a limited adaptation budget merely concentrating probability mass before reliably expressing capability it already possesses. We propose \textbf{entry-state sharpening}: use data-free training \emph{before} TTRL to prepare a general-purpose checkpoint in a state that subsequent label-free adaptation can exploit more efficiently. The idea is not tied to one training recipe; different data-free objectives can move the same base model to different entry states. Across five data-free checkpoints derived from Qwen3-4B and evaluated under an identical 15-step TTRL protocol, entry policy entropy strongly rank-orders endpoint conversion efficiency, a reliability-to-reachability measure (Spearman $ρ=-0.90$; $ρ=-0.99$ after controlling for entry reachability). The contrast across objectives is striking: R-Zero remains diffuse at $3.39$ nats and finishes below the untuned base in 6/6 matched comparisons across MATH, GPQA, and AMC, whereas SPIRAL reaches $0.07$ nats and achieves the highest post-TTRL accuracy on MATH and GPQA despite its self-play stage using no math training data. An in-domain label-free self-distillation intervention further shows that the entry state can be deliberately sharpened. These results motivate treating checkpoint preparation as a \emph{state-control problem}: use data-free training to improve TTRL readiness, with entry entropy as a label-free control signal and reachable capability as the constraint.