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Longer Thoughts, Better Answers: Log-Linear Test-Time Scaling in Reasoning Language Models

Lena Hartmann, Arjun Mehta, Yuki Tanaka · Preprint · August 2026

Abstract

Reasoning language models trained with reinforcement learning write long chains of thought before they answer, and on hard problems their accuracy tends to rise as these chains grow. It remains unclear, however, how reliably accuracy scales with the amount of thinking, and whether longer reasoning is itself the source of the improvement. We study this question with budget forcing, which caps or extends the thinking phase of a model at inference time without changing its weights, and apply it to three models of different sizes trained under an identical recipe.

On AIME 2024–2025, the accuracy of our largest model rises from 28.3% at a budget of 1K thinking tokens to 52.5% at 8K, gaining about eight points with every doubling of the budget. Forcing models to keep thinking after they would naturally stop adds a further 3.6 points. We argue that test-time scaling in reasoning models follows a log-linear law in reasoning length, and that longer reasoning, rather than additional compute in general, is what drives it.

1. Introduction

Prompting language models to write out intermediate steps improves their performance on multi-step problems [Wei et al., 2022], and sampling several such chains and taking a majority vote improves it further [Wang et al., 2023]. A recent generation of reasoning models goes beyond prompting: trained with reinforcement learning on verifiable rewards, they learn to produce extended traces in which they plan, check intermediate results and revise earlier steps before committing to an answer [DeepSeek-AI, 2025].

These models invite a simple question: if a model is allowed to think for longer, does it answer better, and by how much? Prior work has shown that inference-time compute can substitute for model size on some tasks [Snell et al., 2024], and that a model’s thinking can be lengthened by suppressing its end-of-thinking token [Muennighoff et al., 2025]. What is still missing is a controlled measurement of how accuracy depends on the length of a single reasoning trace, and of whether that dependence holds across scales.

We provide such a measurement. Our contributions are a scaling study of accuracy against thinking budget for three model sizes, an intervention that extends reasoning beyond the point where models would naturally stop, and a log-linear fit that predicts accuracy at intermediate budgets to within one point.

2. Experimental Setup

We train three decoder-only models with 1.5B, 7B and 32B parameters, initialised from the same family of pretrained checkpoints and fine-tuned with an identical reinforcement learning recipe on 120K competition-style math problems with verifiable answers. All models write their reasoning between dedicated thinking delimiters, followed by a short final answer.

We evaluate on the 60 problems of AIME 2024 and 2025, and report results on MATH-500 in the appendix. For every problem we draw 16 samples at temperature 0.6 and report mean accuracy across samples. We control the length of the thinking phase with two interventions:

Capping: as soon as the trace reaches the budget B, we insert the end-of-thinking delimiter and ask the model to give its final answer.
Extension: if the model emits the delimiter before reaching B, we remove it and append the token “Wait” so that it keeps reasoning, repeating this up to a fixed number of times.

Capping on its own measures how much of its natural reasoning a model can afford, while extension tests whether reasoning past its natural stopping point still helps. Unless stated otherwise, budgets count thinking tokens only and exclude the final answer.

3. Experiments

3.1 Scaling with the Thinking Budget

Table 1 reports accuracy under capping at budgets from 1K to 16K tokens. Accuracy rises monotonically with the budget at every scale, and larger models gain more from each doubling of the thinking budget than smaller ones do.

Budget	1.5B	7B	32B
1K	9.4	18.1	28.3
2K	13.0	24.6	36.7
4K	16.9	30.8	45.0
8K	20.2	36.9	52.5
16K	21.1	38.8	55.8

Between 1K and 8K tokens, the 32B model gains 8.4, 8.3 and 7.5 points with successive doublings; a log-linear fit over this range gives a slope of 8.1 points per doubling (R² = 0.99). The gain from 8K to 16K is smaller, at 3.3 points, which we attribute to the fact that only 18% of traces reach 16K tokens before the model stops on its own.

3.2 Extending Reasoning Past the Natural Stop

Capping cannot tell us whether reasoning past a model’s natural stopping point would still be useful. We therefore fix the budget at 16K and apply extension, appending “Wait” up to two times whenever a model ends its trace early. Forcing the models to keep thinking improves accuracy at every scale, indicating that longer reasoning itself, rather than the budget cap, drives the gains.

Model	Natural stop	+ Wait ×1	+ Wait ×2
1.5B	21.1	22.0	22.4
7B	38.8	40.9	41.7
32B	55.8	58.2	59.4

With two extensions the 32B model reaches 59.4%, 3.6 points above its natural stop. Inspecting the problems whose outcome changes, we find that in about 70% of cases the model revisits an earlier answer and corrects it, while in the remaining cases it abandons its first approach and works through a new one.

4. Discussion

Across three scales, accuracy grows log-linearly with the length of the reasoning trace up to 8K tokens, and keeps improving when models are made to think past their natural stopping point. We take this as evidence that reasoning length is the main lever of test-time scaling in these models, and that budgets should be set per problem rather than fixed in advance.

An alternative explanation is that the gains reflect additional compute rather than longer reasoning per se. In a preliminary comparison, majority voting over four independent 4K-token traces reaches 56.7% with the 32B model, slightly above a single 16K-token trace (55.8%) at the same total budget. We have not yet run this comparison systematically across budgets and model scales.

Our analysis also treats the visible trace as a faithful record of the model’s computation, which prior work suggests it may not always be [Lanham et al., 2023]. Compute-matched comparisons with parallel sampling, and truncation tests that check when the final answer is settled, would help separate these accounts; we leave both to future work.

5. Conclusion

We measured how the accuracy of reasoning language models depends on the length of their thinking. Across three model sizes, accuracy grows log-linearly with the thinking budget up to 8K tokens, and extending reasoning past the natural stopping point brings further gains. These results suggest that test-time scaling can be predicted from reasoning length alone, and they motivate adaptive budgets that let a model think for as long as each problem requires.

Used by Researchers at

R1Keyword filter258

looped transformerrecurrent depthweight-tieduniversal transformer

R2Keyword filter312

adaptive computation timeearly exitdynamic depthmixture-of-recursions

R3Semantic search600

reusing one transformer block across depth to get recurrent depth

R4Semantic search400

reasoning in latent space by looping a block instead of writing CoT

R5Semantic search300

what looped transformers can express, and why recurrence helps

Merge & dedupe

AI screeningJev320 / 1,412

Inclusion probabilityCut-off 0.5

00.51.0

TitleProb

1Thinking Deeper, Not Longer: Depth-Recurrent Transformers0.98

2One Model to Loop Them All: Elastic Latent Recurrence0.97

3Closing the Loop: Training Recipes for Looped LMs0.97

4What Makes Recurrence Effective in Looped LMs?0.96

AI classification

Architecture design space124

TitleProb

1Hyperloop Transformers0.96

2Scaling Latent Reasoning via Looped Language Models0.95

3Relaxed Recursive Transformers0.93

4Mixture-of-Recursions: Dynamic Recursive Depths0.91

Adaptive test-time compute42

TitleProb

1Mixture-of-Recursions: Dynamic Recursive Depths0.97

2Think-at-Hard: Dynamic Looped Transformers0.95

3AdaPonderLM: Gated Pondering Language Models0.93

4Enabling Dynamic Computation in Looped LMs0.91

Mechanisms & interpretability40

TitleProb

1Think Shallow, Solve Deep: Controlling Recurrent Dynamics0.94

2Loop as a Bridge: Representation Space and Language0.92

3Looped Transformers under the Jacobian Lens0.90

4What Makes Recurrence Effective in Looped LMs?0.88

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4. Discussion

4. 讨论

另一种解释是，这些收益反映的是额外的算力，而不是更长的推理本身。在一项初步比较中，32B 模型对四条相互独立的 4K token 推理做多数投票，准确率达到 56.7%。

4. 考察

別の説明として、これらの向上は長い推論そのものではなく、追加の計算量を反映しているとも考えられる。予備的な比較では、32B モデルで互いに独立した 4K トークンの推論 4 本の多数決が 56.7% に達した。

4. 논의

다른 설명으로, 이러한 향상은 더 긴 추론 자체가 아니라 추가 연산을 반영한 것일 수도 있다. 예비 비교에서 32B 모델로 서로 독립적인 4K 토큰 추론 네 개에 다수결을 적용하자 56.7%에 이르렀다.

Original

The gains may come from extra compute rather than longer reasoning.1

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