Token Budget Saturation and Mechanistic Early-Detection of Reasoning Non-Convergence in Chain-of-Thought Models

Chain-of-thought reasoning in large language models carries a substantial inference cost, yet the relationship between allocated thinking budget and downstream accuracy remains poorly characterized. We present two interconnected studies on DeepSeek-R1-Distill-Qwen-7B across three mathematics benchmarks (GSM8K, MATH-500, AIME 1983-2024). In the first study, we adapt the budget-forcing mechanism of Muennighoff et al. to constrain the model's thinking tokens and measure accuracy as a function of budget. We find that GSM8K and MATH-500 reach 95% of their uncapped accuracy at just 256 tokens of thinking, while AIME exhibits a pronounced bimodal pattern: 56.5% of generations converge naturally with 96.5% accuracy, while the remaining 43.5% never terminate their reasoning chain even at a 10,000-token ceiling and achieve only 11.5% accuracy. This bimodality persists across all difficulty levels and is only weakly predicted by problem difficulty ($r^2 \approx 0.186$). In the second study, we probe whether convergence failure is detectable from internal representations before behavioral failure manifests. Sweeping all 28 layers at token 150, we find layer 20 yields the strongest signal (AUC $0.608 \pm 0.080$). A nine-position checkpoint sweep (50-300 tokens) shows activation probes outperforming behavioral baselines at 8 of 9 positions (mean $\Delta$AUC $= +0.035$), a consistent directional advantage that $n=200$ is underpowered to confirm statistically at individual checkpoints. A contamination check on post-cutoff AIME 2025 problems confirms that the convergence-correctness relationship holds on unseen data (converged accuracy 100%, non-converged accuracy 0%), ruling out memorization as the primary driver. These findings suggest that reasoning models encode convergence fate in their internal state well before it becomes externally observable, motivating mechanistically-grounded early-exit strategies for inference cost reduction.