Abstract
SFT-then-RLVR is widely used for post-training reasoning models, but why this specific ordering, and why RLVR-only stalls at cold start, have lacked a unifying theoretical account. We provide that account under a unified loss family J_Q using the Tsallis q-logarithm. J_Q is a single-parameter family that interpolates between RLVR (at q=0, the \textit{exploitation pole}) and the log-marginal-likelihood over latent trajectories (at q=1, the \textit{density-estimation pole}), under which the standard pipeline corresponds to a stepwise q=1 → 0 schedule. All members share the same per-example gradient direction, differing only by a per-instance amplification P_θ^-q that reweights each instance independently of the learning rate. Under gradient flow analysis, we show that the exploitation pole requires Ω(1/p₀) time to escape cold start but is robust to label noise, while the density-estimation pole escapes in Θ(log(1/p₀)) but memorizes label noise. This separation explains how SFT (q=1) first moves the model out of the cold-start regime, followed by the more robust RLVR (q=0), under the SFT-then-RLVR paradigm. We further derive two Monte Carlo estimators that directly optimize fixed-q on the J_Q continuum, without annotated rationales: Gradient-Amplified RL (GARL) and Posterior-Attenuated Fine-Tuning (PAFT), with shared bias O(q/M P_θ^q) but different variance and stability properties. On FinQA, HotPotQA, and MuSiQue, GARL at sufficiently high q substantially mitigates cold-start stalling, escaping cold start where GRPO fails entirely. In warm start, GARL at low q dominates FinQA where training is stable; on HotPotQA and MuSiQue, GARL destabilizes and PAFT at q=0.75 remains stable, reaching 47.9 \texttt{m@16} on HotPotQA (+13.9 over GRPO).