# Calibrated e-CUSUM Decoding for Quantized Reasoning Models: Why Token Log-Probability Is the Wrong Observable for Decoding Monitors

arXiv: https://arxiv.org/abs/2607.11317 · HTML: https://mlsanity.com/paper/2607.11317/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Test-time compute and efficient thinking, one of 17 directions in LLM reasoning.
- Builds on 22 of its 27 references that are in the map.
- First posted on arXiv 13 Jul 2026.

## Placed in

- LLM reasoning / [Test-time compute and efficient thinking](https://mlsanity.com/lr/test-time-compute-and-efficient-thinking/)

## Builds on (most cited first, 22 of 22)

- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [Self-Consistency Improves Chain of Thought Reasoning in Language Models](https://mlsanity.com/paper/2203.11171/) (2022)
- [Let's Verify Step by Step](https://mlsanity.com/paper/2305.20050/) (2023)
- [QLoRA: Efficient Finetuning of Quantized LLMs](https://mlsanity.com/paper/2305.14314/) (2023)
- [The Curious Case of Neural Text Degeneration](https://mlsanity.com/paper/1904.09751/) (2019)
- [Deep Think with Confidence](https://mlsanity.com/paper/2508.15260/) (2025)
- [The Unreasonable Effectiveness of Entropy Minimization in LLM Reasoning](https://mlsanity.com/paper/2505.15134/) (2025)
- [Thoughts Are All Over the Place: On the Underthinking of o1-Like LLMs](https://mlsanity.com/paper/2501.18585/) (2025)
- [Game-theoretic statistics and safe anytime-valid inference](https://mlsanity.com/paper/2210.01948/) (2022)
- [Turning Up the Heat: Min-p Sampling for Creative and Coherent LLM Outputs](https://mlsanity.com/paper/2407.01082/) (2024)
- [Quantization Hurts Reasoning? An Empirical Study on Quantized Reasoning Models](https://mlsanity.com/paper/2504.04823/) (2025)
- [Repetition In Repetition Out: Towards Understanding Neural Text Degeneration from the Data Perspective](https://mlsanity.com/paper/2310.10226/) (2023)
- [E-detectors: a nonparametric framework for sequential change detection](https://mlsanity.com/paper/2203.03532/) (2022)
- [Step Back to Leap Forward: Self-Backtracking for Boosting Reasoning of Language Models](https://mlsanity.com/paper/2502.04404/) (2025)
- [Self-Correction Bench: Uncovering and Addressing the Self-Correction Blind Spot in Large Language Models](https://mlsanity.com/paper/2507.02778/) (2025)
- [WATCH: Adaptive Monitoring for AI Deployments via Weighted-Conformal Martingales](https://mlsanity.com/paper/2505.04608/) (2025)
- [Penalty Decoding: Well Suppress the Self-Reinforcement Effect in Open-Ended Text Generation](https://mlsanity.com/paper/2310.14971/) (2023)
- [Adaptive Conformal Inference by Betting](https://mlsanity.com/paper/2412.19318/) (2024)
- [Quantized Reasoning Models Think They Need to Think Longer, but They Do Not](https://mlsanity.com/paper/2606.00206/) (2026)
- [Quantization Meets Reasoning: Exploring and Mitigating Degradation of Low-Bit LLMs in Mathematical Reasoning](https://mlsanity.com/paper/2505.11574/) (2025)
- [Mitigating Overthinking in Large Reasoning Language Models via Reasoning Path Deviation Monitoring](https://mlsanity.com/paper/2603.14251/) (2026)
- [Circular Reasoning: Understanding Self-Reinforcing Loops in Large Reasoning Models](https://mlsanity.com/paper/2601.05693/) (2026)

## Abstract (arXiv)

Low-bit quantization makes small reasoning models inexpensive to deploy but can degrade their chains of thought. This motivates decoder-side monitors that intervene when generation becomes unreliable. We show that a natural candidate, the centered token log-probability increment log p(wt)+Ht, is the wrong observable for this purpose. Under the model's own sampling law it is a mean-zero martingale by construction, so it measures sampling self-consistency rather than trajectory health and is nearly silent during confident repetition, where both log p(wt) and entropy are close to zero. We introduce a training-free decoding controller that combines (i) a degeneration-aware alarm score fusing token uncertainty with explicit verbatim repetition and (ii) a calibrated e-process-inspired sequential detector. The raw product process is Ville-valid under a conditional-mean null, while the deployed CUSUM-floored statistic is treated as an empirical change detector because the score is history-dependent and autocorrelated. On GSM8K with DeepSeek-R1-Distill-Qwen-1.5B in FP16 and INT4, calibration turns a monitor that fires on 93--95% of generations into a selective detector of failing traces (φ ≈ 0.3, precision ≈ 0.6 against a 0.38 base rate). In this pilot, the controller reduces measured verbatim-degeneration signals and yields a positive but statistically inconclusive INT4 accuracy change from 63% to 69% (paired McNemar p=0.18, n=100), at a 28% token-budget cost. We also find that non-termination, rather than looping, is the dominant failure mode on GSM8K. The main contribution is methodological: an explanation of why centered token log-probability is inadequate for decoder monitoring and a calibrated, cautiously evaluated replacement.