Adapting reasoning models to new tasks during post-training with only output-level supervision stalls under reinforcement learning from verifiable rewards (RLVR) when the initial success probability $p_0$ is small. Using the Tsallis $q$-logarithm, we define a loss family $J_Q$ that interpolates between RLVR (at $q{=}0$, the exploitation pole) and the log-marginal-likelihood over latent trajectories (at $q{=}1$, the density-estimation pole). All members share the same per-example gradient direction, differing only by a scalar amplification $P_{θ^{-q}}$ that reweights each instance independently of the learning rate. This amplification is the mechanism that addresses cold-start stalling: under gradient flow, the exploitation pole requires $Ω(\frac{1}{p_0})$ time to escape cold start, while the density-estimation pole escapes in $Θ\big(\log(\frac{1}{p_0})\big)$; intermediate $q$ trades escape speed against noise memorization. Because $P_θ$ is intractable, we derive two Monte Carlo estimators from the two factorizations of the gradient: Gradient-Amplified RL (GARL) samples from the prior and amplifies the RL gradient, and Posterior-Attenuated Fine-Tuning (PAFT) importance-resamples from the posterior and runs standard SFT. Both have bias $O\big(\frac{q}{M P_θ^{q+1}}\big)$; GARL has lower variance, PAFT has semantically coherent gradients. On FinQA, HotPotQA, and MuSiQue, GARL at $q{=}0.75$ 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 during training, and PAFT at $q{=}0.75$ provides stable gradients (best overall on HotPotQA at 47.9 maj@16, $+14.4$ over GRPO).

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