Cross Domain Test Time Scaling: Scale Knowledge and Reasoning on Cross Domains

1 Introduction

Recently, test-time scaling (TTS) has demonstrated its power in large language models (LLMs) [Snell et al., 2025]. If pretrained model size can be traded off for additional inference compute, this would enable the deployment of smaller ondevice models in place of datacenter-scale LLMs. The most mainstream TTS methods generally fall into two categories: sequential scaling and parallel scaling. Sequential scaling iteratively updates intermediate states, iterative refinement, and revises the answer. The most widely used sequential scaling approach includes budget forcing, self-refinement, and CoT [Wei et al., 2022; Kamoi et al., 2024]. Parallel scaling enhances the reasoning abilities of LLMs by generating multiple answers and selecting the best one. For parallel scaling, the most widely used approach is repeated sampling, including techniques like best-of-n sampling, majority voting, and self-consistency [Inoue et al., 2026; Li et al., 2025]. Recently, some researchers have also tried to combine sequential scaling and parallel scaling. This method, called hybrid scaling, exploits the complementary benefits of both parallel and sequential approaches.[Inoue et al., 2026]

However, these existing TTS technologies primarily rely on the mathematical and coding domains due to their inherent ease of validation and reasoning-intensive nature. The TTS capabilities of LRMs in other domains, such as medical and general scientific fields, remain largely underexplored. Recent research has shown that s1[Muennighoff et al., 2025] is ineffective in medical domains[Huang et al., 2025]. Furthermore, the general reasoning abilities acquired by R1-distilled models do not transfer effectively to the medical domain through either Supervised Fine-Tuning (SFT) or Reinforcement Learning [Oh et al., 2025]. Other studies also indicate that SFT on mathematical data can even lead to a performance decrease in non-reasoning domains[Huan et al., 2025].

Inspired by [Wu et al., 2025b], we can divided a reasoning task into two parts, reasoning depth and reasoning width, the former represent the reasoning, and the later represents the inner knowledge of LRMs. Furthermore, during the thinking process, if a path stops providing new information, we should accelerate the pace to reach the conclusion more quickly.

Our contributions are threefold:

• We introduce a novel unified test-time scaling (TTS) method that demonstrates robust effectiveness across both reasoning-intensive and knowledge-intensive domains.
• We present a method to successfully mitigate the previously observed ineffectiveness of TTS in knowledgeintensive tasks, thereby broadening the applicability of test-time scaling for large reasoning models.
• We propose a cross-domain reasoning method based on test-time scaling. Unlike most training-based approaches, ours is training-free and significantly more resource-efficient.

2 RelatedWorks

2.1 Large ReasoningModels

The field of Large Reasoning Models (LRMs) has witnessed significant progress in recent years. Notably, models such as the DeepSeek-R1 series [Guo et al., 2025] and the OpenAIO1 series [Jaech et al., 2024] have demonstrated strong performance in domains like mathematics and coding. However, the development of LRMs in other critical areas, such as medicine and finance, remains relatively underexplored. In the medical domain, recent advancements have shown promise. For instance, Huatuo-GPT-O1 [Chen et al., 2024a], MedReason [Wu et al., 2025a], and Med-R1 [Lai et al., 2025] have made significant strides in enhancing medical reasoning capabilities. Also, in the financial domain, models like Fin-R1 [Liu et al., 2025] have been developed to address the unique challenges of financial reasoning. Fin-R1 incorporates domain-specific knowledge graphs and reinforcement learning techniques to improve its ability to analyze financial data and make accurate predictions. However, despite these advancements, LRMs in domains other than mathematics and coding still face significant challenges and opportunities for further development.

2.2 Cross Domain Reasoning

Large Reasoning Models have shown their promising potential. However, open research has been narrowly focused on math and code domains. Recently, the academic community has noticed this lack. Cross Domain Reasoning has get advancement. Reasoning360 [Cheng et al., 2026] introduces GURU-7B and GURU-32B models, achieving state-of-theart performance for open models trained with publicly available data. Similarly, Open-Reasoner-Zero [Hu et al., 2026] scale reinforcement learning directly on base large language models for reasoning tasks. Also, General-Reasoner [Ma et al., 2026] developed a generative model-based verifier for robust answer assessment within a Zero RL training framework based on a verifiable all-domain reasoning dataset. CrossThink [Akter et al., 2026] systematically incorporates multidomain and multi-format data with rule-based rewards, leading to LLMs that outperform the base model by 8.55% to 13.36% across diverse reasoning benchmarks and generate more efficient, concise, correct answers. Crossing the Reward Bridge [Su et al., 2025] employs a generative, model-based soft reward method to extend RLVR to more diverse, broader domains. However, research on cross-domain reasoning is still in its nascent stages, and most methods remain confined to reinforcement learning.

2.3 Test Time Scaling for LLMs

Test-time scaling (TTS) refers to techniques that adjust the computational effort exerted by a model during inference to improve performance. This paradigm is particularly relevant for LRMs, where a trade-off between model size and inference compute can enable more flexible deployment. TTS methods aim to enhance the quality of the output by allocating additional computational budget at inference time, rather than solely relying on a fixed, pre-trained model. Test Time Scaling can be generally divided into three categories: sequential scaling, parallel scaling, and hybrid scaling.

Sequential scaling methods [Muennighoff et al., 2025] enhance reasoning depth. These methods iteratively update intermediate states, often by prompting the model to engage in more extensive thinking. For instance, s1 [Muennighoff et al., 2025] introduces test-time scaling through budget forcing, which controls computational effort during inference by appending "Wait" tokens to extend thinking or by forcefully terminating a response. Similarly, alphaone [Zhang et al., 2025a] employs a hyperparameter α to adjust the balance between slow thinking and fast reasoning processes, thereby achieving strong performance.

In contrast, parallel scaling can expand the reasoning width. Unlike sequential scaling, parallel scaling will generate multiple outputs for one sample, and then, by techniques such as best of N [Snell et al., 2025], aggregating them into a final answer[Zhang et al., 2025b; Zeng et al., 2025].

Recently, hybrid scaling has also emerged as a promising approach, combining the strengths of sequential scaling and parallel scaling[Bi et al., 2025; Inoue et al., 2026]. Some methods employ multi-agent frameworks to further enhance the robustness of scaling [Wang et al., 2025; Chen et al., 2024b]. However, most of these TTS methods currently focus primarily on the mathematical and coding domains, neglecting their potential applications in other domains.

3 Method

Our framework is composed of three primary parts. First, it utilizes a Conformal Prediction-based cold-start strategy to determine the necessity of parallel scaling at the start of LLM processing. Second, a Thinking Progress Vector (TPV) is trained on a calibration dataset to identify a progress vector that defines the acceleration direction within the reasoning trajectory. Third, we implement a dynamic speed adjustment approach based on information gain, which adaptively modulates the reasoning pace by quantifying the marginal utility of each incremental step in the model's output.

3.1 Conformal Prediction Cold Start

To establish the cold-start strategy for reasoning, we employ Conformal Prediction (CP) [Angelopoulos and Bates, 2023], a rigorous statistical framework designed to quantify the uncertainty of model predictions without requiring specific distributional assumptions. During the calibration phase, we utilize a hold-out calibration dataset Dcal = {(xi, yi)}ni=1 . For each instance, the LLM generates a probability distribution over the label space Y = {A,B,C,D}. Let ˆ f(xi)y denote the softmax probability assigned to label y. We define the conformity score si as the probability of the true label:

Given a pre-defined error budget α ∈ (0, 1), we calculate the threshold ˆ q as the α-quantile of the scores in Dcal:

The theoretical strength of CP lies in its marginal coverage guarantee, which ensures that for a new test point xn+1, the true label yn+1 is contained in the prediction set C(x) with a probability of at least 1-α:

During inference, for any input query X, the prediction set C(X) is constructed by collecting all options whose softmax probabilities exceed the calibrated threshold ˆ q:

The cardinality of C(X) serves as a proxy for the model's epistemic uncertainty: When |C(X)| = 0, it indicates that the questions in the dataset are of high difficulty or logical complexity for the model. Such data is considered reasoningintensive and necessitates deep deliberation. If |C(X)| > 1, it suggests that the model exhibits significant uncertainty regarding the answers, often due to knowledge ambiguity or internal conflicts. In this scenario, we employ parallel expansion strategy. Finally, when |C(X)| = 1, it signifies that the answers are generally deterministic and well-represented within the model's parameters. Under these conditions, the model operates without adjustments to its reasoning progression.

3.2 TPV Training

To capture the internal progression of reasoning, we first construct a calibration training set Q. For each problem q ∈ Q, we prompt the base reasoning model to generate multiple independent trajectories. For a reasoning chain containing N tokens, we extract the sequence of hidden states H = {h1, h2, . . . , hN} from the final layer. We define the relative reasoning progress pj for each token as its normalized position within the sequence:

This results in a dataset of pairs D = {(hj, pj)}∑ N j=1 mapping internal states to progress values. Empirical evidence suggests that the reasoning progress of LLMs is linearly encoded within the hidden state space.[Eisenstadt et al., 2025] Consequently, we define the TPV θ as the weight of a linear predictor, optimized by minimizing the Mean Squared Error (MSE):

where θ ∈ Rd represents the TPV. Once the Thinking Progress Vector (TPV) is obtained, we can manually modify the model's hidden states during the inference stage to control the reasoning progress. The formula is:

where α is the intervention strength, hα represents the adjusted hidden states, h represents the original hidden states of the model, and θ is the Thinking Progress Vector.

3.3 Dynamical Thinking Path Adjusting

Building upon the TPV framework, we propose a dynamic intervention mechanism that adjusts the intervention intensity α based on the Information Gain (∆I) of different reasoning stages. Following the InfoGain framework, we quantify the extent to which each reasoning step si within a trajectory S = {s1, s2, . . . , st} reduces the uncertainty regarding the final answer A∗ . The Information Gain ∆Ii is defined as the reduction in the Perplexity (PPL) of the correct answer:

To establish a robust mapping, we first partition the normalized progress p ∈ [0, 1] into multiple discrete intervals(accroding to the previous reasoning steps). Using a calibration set, we generate reasoning trajectories and calculate the average Information Gain ∆Iavg for each interval. This allowed us to identify low-gain stages where the model is prone to overthinking or redundancy. Based on these calibration results, we pre-define a mapping function α(p) that assigns an intervention strength to each progress stage:

During the inference stage, the model only needs to compute the current progress value p = θT h + b at each step. The corresponding intervention strength α is then retrieved from the pre-defined mapping based on the current interval.

4 Experiment

To evaluate our method's effectiveness, we conducted several experiments across different domains, including medical, mathematical, and scientific. For the purpose of a straightforward evaluation, we specifically examined queries with definite answers, such as those found in medical questionanswering or multiple-choice problems, and fill-in-the-blank mathematical problems.

4.1 Settings

Datasets

We selected three domains for our evaluation: medical, as a typical knowledge-intensive domain; mathematics, as a reasoning-intensive domain; and science, chosen as a compromise option that have both knowledge-intensive and reasoning-intensive features. For the medical domain, we evaluated on five medical question-answering benchmarks. Specifically, we used MedQA-USMLE (MedQA) [Jin et al., 2021], the 4-Options (MedBullet (4)) and 5Options (MedBullet (5)) splits from the MedBullets platform (MedBullet) [Chen et al., 2025], Humanity's Last Exam (Med) [Kalinin and others, 2026], and MedXpertQA [Zuo et al., 2025]. All of these are single or multiple-choice QA problems, chosen for their verifiability and high quality. For multimodal tasks, we performed evaluations using medical datasets that feature easily verifiable answers. Specifically, we utilized MedXpertQA(MM) for our experiments. For the mathematics domain, we evaluated on four math datasets: AIME 2024 (AIME24) [Codeforces, ], AMC23 [AIMO, 2024], and MATH500[Hendrycks et al., 2021]. For the science domain, we employed GPQA-diamond [Rein et al., 2024].

Models

We conducted experiments using six large reasoning models to validate the effectiveness of our method. For knowledgeintensive problems like those in medicine, large reasoning models need to possess a certain level of medical knowledge. Therefore, for medical problems, we selected a medicaldomain-specific large model, HuatuoGPT-O1 [Chen et al., 2024a], a cross-domain large model, GURU [Cheng et al., 2026], and two general-purpose large models, Deepseek Distill Qwen [Guo et al., 2025] and Qwen3 [Yang et al., 2025]. For mathematics and science problems, general-purpose large models typically have the requisite knowledge, especially for mathematical problems. Thus, for these two domains, we employed a cross-domain large model, GURU, two sizes of the Deepseek Distill Qwen model, and a general-purpose large model, Qwen3. For multimodal benchmarks, we test our method on Qwen2.5-VL-7B-Instruct [Bai et al., 2025].

Baselines

For language tasks, to validate the effectiveness of our method, we compared it with four baseline approaches. The first, original output, represents the model's Pass@1 output without employing any test-time scaling methods. The second, S1 [Muennighoff et al., 2025], forces the model to continue thinking, refining, and correcting its reasoning by inserting a "wait" token. The third, alphaone [Zhang et al., 2025a], introduces a unified adjustable α-moment to regulate the model's processing speed. The fourth is a parallel sampling method, shortest majority vote strategy(SMV) [Zeng et al., 2025], which addresses the model's overthinking problem by selecting answers with shorter reasoning paths but the highest vote count. For multimodal tasks, most of the TTS methods cannot be transferred directly to align with the multimodal inference process, so we chose zero-shot and CoT as baselines.

4.2 Main Result

Medical Domain

Table 1 presents the results of our method compared to baseline approaches, demonstrating that our proposed crossdomain test-time scaling method outperforms most baselines.

For example, in experiments with the M1 7B model on the more reasoning-intensive HLE (Med) task, s1 and AlphaOne perform better than the short majority vote (SMV). Our method is adaptive and opts for more efficiency and precision for these reasoning-intensive tasks, outperforming the short majority vote by 2.43%. As shown in 3, in multimodal tasks, our method is still effective.

Furthermore, we found that our method is more effective on models that tend to generate extensive reasoning steps, such as M1. When the model fails to obtain useful information during reasoning, our approach can accelerate the reasoning process and enable the model to reach a conclusion more quickly. This prevents the model from repeatedly overturning its own conclusions, reduces token consumption by approximately 50%, and achieves higher pass@1 accuracy.

Math and Science Domain

For the math domain, which is highly reasoning-intensive, as shown in Table 2, our method effectively improves accuracy while reducing token consumption. Experimental results demonstrate that deep reasoning approaches, such as S1 and AlphaOne, remain effective for mathematics-related problems that prioritize logical inference; they guide the model into deeper cognitive processing. For example, on capacitylimited models such as DeepSeek-R1 distilled Qwen-1.5B, the performance gains from test-time scaling are particularly obvious, with our method achieving up to about 12% improvement. Compared with S1 and AlphaOne, our method not only maintains comparable accuracy but also significantly reduces the number of tokens required. Specifically, our results show an average reduction of approximately 30% in token consumption. On benchmarks like scientific problems, where reasoning and knowledge components are more balanced, the advantages of our method are even more pronounced. Our approach achieved the highest accuracy across all three models while saving a substantial number of inference tokens.

4.3 Ablation Studies

Study of the effectiveness of cold start strategy

To validate the effectiveness of the parallel expansion component in our method, we conducted ablation experiments on medical domain, which is a typical knowledge-intensive domain on three models: M1, HuatuoGPT-O1, and DeepseekR1 Distill Qwen 7B. As shown in Figure 3, without the cold start mechanism, the Pass@1 of all three models decreased on most benchmarks, particularly on knowledge-intensive tasks such as MedQA, MedBullet(4), and MedBullet(5), which demonstrates the necessity of parallel scaling, especially for knowledge-intensive tasks such as medicine.

To validate the effectiveness of the slow thinking adjustment component of our method, we conducted ablation experiments on three models: DeepSeek-R1 Distill Qwen 1.5B, DeepSeek-R1 Distill Qwen 7B, and Qwen3 8B, within the mathematics domain. As shown in Figure 4, without the slow thinking adjustment, the Pass@1 of all three models decreased across all benchmarks. These results have shown the necessity of the effectiveness of our cold start strategy.

Study of the effect of information gain-based thinking adjusting

We conducted ablation experiments on the Information Gainbased thinking path, adjusting part of our framework. The ablation experiment was done using the DeepSeek-R1-DistillQwen-7B model across benchmark datasets in three domains: MedQA (medicine domain), AIME24 (mathematics domain), and GPQA (science domain). As shown in Table 4, the results demonstrate the effectiveness of information gain-based thinking. Adjusting this mechanism leads to a performance decline in both pass@1 accuracy and token consumption.

We further observe that the Information Gain for knowledge-oriented tasks is lower than that for tasks requiring complex reasoning. Taking evaluations in the medical domain as an example, we find that the Information Gain measured on MedQA is approximately 0.15, while on MedXpertQA it can reach 0.3. This result suggests that for knowledge-intensive tasks, reasoning steps do not bring about significant Information Gain to help the model arrive at the correct answer. This may also explain why the reasoning process does not substantially improve the answer accuracy of large language models on knowledge-based tasks.

5 Conclusion

In this paper, we propose a cross-domain test-time scaling (TTS) framework comprising a conformal prediction-based cold-start strategy and information gain-based reasoning path adjustment. To be more specific, we use conformal prediction to set the starting point for test-time scaling, then leverage internal progress vectors to dynamically adjust the model's reasoning speed based on the effectiveness of each step in the inference process. We conducted experiments in three representative domains, and both experimental results and ablation studies demonstrate our method's overall effectiveness and the contribution of its key components. We hope our work inspires future research on cross-domain reasoning in LLMs and LVLMs, moving past the limitations of traditional reasoning-intensive and knowledge-intensive tasks.

Acknowledgements

This research was supported by funding from the Hong Kong RGC General Research Fund (Grant Nos. 15221123, 15216424, and 15211525), and the Hong Kong PolyU Internal Research Fund (Grant Nos. P0058468 and P0056171). This research was also supported by funding from the Guangdong Basic and Applied Basic Research Foundation (Grant No. 2026A1515010184).