The prevailing post-training paradigm for Large Reasoning Models (LRMs)—Supervised Fine-Tuning (SFT) followed by Reinforcement Learning (RL)—suffers from an intrinsic optimization mismatch: the rigid likelihood maximization in SFT induces distributional collapse, thereby exhausting the exploration space necessary for subsequent RL. Motivated by the Gibbs optimum of KL-regularized RL, we derive a token-level variational surrogate that makes the SFT target structurally compatible with the subsequent RL stage, and propose Gibbs Initialization with Finite Temperature (GIFT). Standard SFT emerges as a degenerate zero-temperature limit of this surrogate, while a finite temperature preserves structural diversity. Our experiments demonstrate that GIFT outperforms standard SFT and surpasses competitive baselines when utilized for RL initialization. Our code is available at https://github.com/zzy1127/GIFT.

Synthetic math data is widely used to scale LLM training, but its correctness is not guaranteed. We introduce MathDebugger, the first text-only, type-aware benchmark for jointly detecting *and* classifying errors in synthetic mathematical questions and answers. MathDebugger contains 2,000 correct questions, 2,000 erroneous questions (balanced across four fine-grained error types), and 2,000 annotated answers (610 erroneous, across three error types), each annotated with both a correctness label and an error-type label, with per-error-type Fleiss’ 𝜅 ranging from 0.69 to 0.88. On 14 SOTA LLMs (6 closed-source, 8 open-source, 1.5B–72B) and 3 process reward models, even advanced systems such as GPT-o3 and DeepSeek-R1 remain far from saturating the benchmark, particularly on fine-grained error-type detection; we further surface a *solving-vs-verifying* gap that causes math-tuned and long-chain-reasoning models to trail their general-purpose siblings. Error-type annotations also provide actionable supervision for correction, with gains that are significant under paired bootstrap (p<0.01). Code and data: [https://anonymous.4open.science/r/MathDebugger-7C4A/README.md](https://anonymous.4open.science/r/MathDebugger-7C4A/README.md)