Paper deep dive
Fusion Training for Mathematical Generalization in Large Language Models
Congfeng Cao, Pengyu Zhang, Jelke Bloem
Intelligence
Status: not_run | Model: - | Prompt: - | Confidence: 0%
Entities (0)
Relation Signals (0)
No relation signals yet.
Cypher Suggestions (0)
No Cypher suggestions yet.
Abstract
Abstract:Thinking Mode Fusion (TMF) enables large language models to support both concise responses and long-form reasoning by unifying a non-thinking mode and a thinking mode within a single model. However, its training dynamics, including the \emph{data ratio} and \emph{training schedule} between the two modes, remain underexplored. In this work, we present a systematic study of TMF by analyzing the effects of the training schedule and data ratio between thinking and non-thinking modes. Focusing on mathematical problem solving, we construct a benchmark with multiple thinking-to-non-thinking data ratios and three training schedules. Our results reveal an asymmetric interaction between the two modes: increasing the ratio of non-thinking supervision reduces the accuracy of the thinking mode. We further show that different training schedules modulate this trade-off and that the optimal schedule depends on the data ratio. Finally, we quantify a negative correlation between non-thinking and thinking mode supervision, highlighting an inherent tension between these two modes. These findings provide practical guidance for designing effective TMF training settings. All code and data are released to support further research at: \href{this https URL}{\textbf{Fusion Bench}}.
Tags
Links
- Source: https://arxiv.org/abs/2608.09893v1
- Canonical: https://arxiv.org/abs/2608.09893v1
Trouble viewing inline? Open PDF directly →
Full Text
50,999 characters extracted from source content.
Expand or collapse full text
Fusion Training for Mathematical Generalization in Large Language Models Congfeng Cao 1 , Pengyu Zhang 2 , Jelke Bloem 1 1 Institute for Logic, Language and Computation, University of Amsterdam 2 INDE Lab, University of Amsterdam, Abstract Thinking Mode Fusion (TMF) enables large language models to support both concise re- sponses and long-form reasoning by unifying a non-thinking mode and a thinking mode within a single model. However, its training dynamics, including the data ratio and training schedule between the two modes, remain underexplored. In this work, we present a systematic study of TMF by analyzing the effects of the training schedule and data ratio between thinking and non-thinking modes. Focusing on mathemati- cal problem solving, we construct a benchmark with multiple thinking-to-non-thinking data ra- tios and three training schedules. Our results reveal an asymmetric interaction between the two modes: increasing the ratio of non-thinking supervision reduces the accuracy of the think- ing mode. We further show that different train- ing schedules modulate this trade-off and that the optimal schedule depends on the data ra- tio. Finally, we quantify a negative correlation between non-thinking and thinking mode su- pervision, highlighting an inherent tension be- tween these two modes. These findings provide practical guidance for designing effective TMF training settings. All code and data are released to support further research at: Fusion Bench. 1 Introduction Large language models (LLMs) have made sub- stantial progress in long-form reasoning and have been increasingly applied to complex tasks via ex- tended chains of thought (Wei et al., 2022; Kojima et al., 2022; Wang et al., 2023). However, many real-world queries are comparatively simple and do not require long-form reasoning. For such cases, long-form reasoning can be unnecessarily time- consuming and computationally expensive. As a result, existing efforts have explored switch- ing models between simple queries and complex problems (OpenAI, 2024, 2025), or adapting the length of model responses to improve reasoning Thinking Mode Fusion Asymmetric interaction between two modes Thinking dataNon-thinking data Training schedule and data ratio modulate this interaction Data RatioTraining Schedule Figure 1: An overview of the experimental design for analyzing Thinking Mode Fusion under different train- ing orders and data ratios. efficiency (Arora and Zanette, 2025; Kimi et al., 2025; Shen et al., 2025; Wu et al., 2025; Zhang et al., 2025). Qwen3 series models address this limitation by introducing Thinking Mode Fusion (TMF) during post-training (Yang et al., 2025). TMF unifies a thinking mode and a non-thinking mode within a single model, enabling users to con- trol reasoning behavior for both long-form reason- ing and concise responses. TMF is implemented through two different chat templates, i.e., a think- ing chat template and a non-thinking chat template, that support mode selection. This design reduces deployment complexity and inference costs while allowing the model to adjust its reasoning depth under different computational budgets. Despite Qwen3’s strong empirical performance, the mechanisms of TMF and the interaction be- tween the two modes remain underexplored. In par- ticular, it remains unclear how the training sched- ule, i.e., the training order of the two modes, and the data ratio, i.e., the combination of thinking and non-thinking data, affect overall model perfor- arXiv:2608.09893v1 [cs.CL] 10 Aug 2026 mance. This raises a practical question: when both modes are trained within a single model, how do the training schedule and the data ratio affect the model’s performance? To investigate this question, we focus on math- ematics, where problem difficulty varies widely, from short-solution questions to Olympiad-level tasks that require long-form reasoning. As shown in Figure 1, we systematically investigate the in- teraction between the two modes and analyze the effects of TMF under different training schedules and varying ratios of thinking to non-thinking data. To support this study, we construct a benchmark, Fusion Bench, in accordance with the Qwen3 de- sign and consisting of two types of mathematical problems: simple problems for non-thinking mode training and Olympiad-level problems for think- ing mode training under diverse data ratios and training mode combinations. Our results reveal an asymmetric interaction between the two modes: the trade-off between the two modes shows that think- ing mode performance degrades as non-thinking supervision increases. Note that TMF unifies two modes (long-form reasoning and concise response) within a single model for a single task, which is different from multi-task settings such as math and code. Moreover, we show that the training schedule modulates this trade-off and that the optimal sched- ule depends on the data ratio. Finally, we quantify a negative correlation between non-thinking and thinking mode supervision, highlighting an inher- ent tension between these two modes. Our main contributions are as follows: •We conduct a systematic study of training schedules and data ratios in TMF, revealing a trade-off between thinking and non-thinking modes, where increased non-thinking supervi- sion degrades thinking performance, and we quantify a negative correlation between the two modes. • We show that training schedules modulate this trade-off: while the optimal schedule de- pends on the data ratio, the Mix training sched- ule consistently achieves strong performance across both thinking and non-thinking modes when averaged over different ratios. •We construct and release Fusion Bench 1 , a benchmark comprising thinking and non- thinking datasets with varying data ratios and 1 https://anonymous.4open.science/r/Fusion-Bench-2D62 training schedules, enabling controlled and reproducible analysis of TMF. 2 Related Work Thinking Mode Fusion and Hybrid Thinking. Large language models (LLMs) have shown that supervised fine-tuning (SFT) on long chain-of- thought (Wei et al., 2022) traces is able to improve performance on complex tasks. However, enhancing reasoning depth through specialized post-training increases computational costs due to the long reasoning responses. Models such as OpenAI o1 (OpenAI, 2024), DeepSeek- R1 (DeepSeek-AI, 2025), and Gemini (Team, 2025) widely adopt hybrid thinking mechanisms that control whether the model engages in reason- ing, thereby achieving a more efficient and flexi- ble reasoning process. Existing efforts to improve reasoning efficiency focus on reducing the length of model responses (Arora and Zanette, 2025; Kimi et al., 2025; Shen et al., 2025). Wu et al. (2025) merge reasoning and non-reasoning mod- els to reduce output length. AdaptThink proposes teaching reasoning models to choose the optimal thinking mode adaptively based on problem dif- ficulty (Zhang et al., 2025). Qwen3 (Yang et al., 2025) introduced Thinking Mode Fusion (TMF), which allows a single model to support both long- form reasoning and concise responses controlled by the chat format. Wang et al. (2025) further demys- tify hybrid thinking, revealing that current hybrid thinking LLMs only achieve partial mode separa- tion, where reasoning behaviors often leak into the no-think mode. However, the mechanisms of TMF remain underexplored. Data Mixture.Prior studies have shown that data mixture plays a critical role in both pre-training and post-training stages. Ye et al. (2025a) out- lined the quantitative predictability of model per- formance with respect to mixture proportions. Liu et al. (2025) propose RegMix to automatically iden- tify a high-performing data mixture by formulating it as a regression task. Li et al. (2025) frame the selection of data proportions as an optimization problem to minimize validation loss across multi- ple tasks during SFT. While these studies provide insights into domain- level mixing (e.g., balancing math, code, and chat), they lack an examination of the internal interfer- ence between different response formats within the same domain, which is a key focus of our study. Math Datasets.We group math datasets into two categories that align with the thinking (with long reasoning) versus non-thinking (with short solu- tions) modes considered in this paper. On the simple data side, a complementary line of work focuses on datasets that admit short solutions or direct answer supervision, which better aligns with the non-thinking mode. Classic benchmarks such as GSM8K (Cobbe et al., 2021), which con- sists of grade-school math problems, are widely used for instruction tuning and evaluation, and are often employed in short-solution settings to encour- age concise generation. Calc-X (Kadl ˇ cík et al., 2023) is a collection of simple arithmetic-focused math word problems, comprising over300,000 curated samples. Patel et al. (2021) introduced MAWPS, an elementary-level benchmark dataset featuring concise solutions and final answers. In this work, we use GSM8K as the non-thinking mode base dataset, as it covers a diverse range of math problems and is widely adopted. On the long reasoning data side, recent work has released large-scale datasets that explicitly provide long reasoning traces, which are crucial for training and analyzing long-form mathematical reasoning. OpenMathReasoning (Moshkov et al., 2025) is a generated large-scale math reasoning dataset for training large language models (LLMs), containing 3.2million long chain-of-thought (CoT) solutions. OpenMathInstruct-1 (Toshniwal et al., 2024) and OpenMathInstruct-2 (Toshniwal et al., 2025) pro- vide1.8million and14million generated solutions, respectively, covering a wide range of mathemat- ical difficulty levels. We also note curated math reasoning benchmarks such as OMNI-MATH (Gao et al., 2025) and LIMO (Ye et al., 2025b), which are commonly used to evaluate advanced reasoning ca- pabilities. In addition, several datasets emphasize challenging questions paired with chain-of-thought annotations, including Skywork-MathQA (Zeng et al., 2024), which contains2.5million question- answer pairs, and NuminaMath (LI et al., 2024), which consists of860, 000competition-level prob- lems with long-form reasoning. For ease of eval- uation, we choose OpenMathReasoning since it provides problems with exact and verifiable final answers, unlike many other datasets that include proof-based or open-ended reasoning tasks without checkable solutions. In this work, we sample data from the OpenMathReasoning subset generated by DeepSeek-R1 (DeepSeek-AI, 2025) and select ex- amples with a high pass rate (greater than0.96) Table 1: Benchmark statistics for thinking and non- thinking training data under different data ratios. T de- notes thinking mode data and NT denotes non-thinking mode data. We fix the thinking mode subset size (N T = 1500) and vary the non-thinking subset size (N NT ) accordingly. Mean denotes the mean length of the reasoning traces. ModeRatio (T : NT) Count Mean MinMax Non-thinking 4 : 13756713301,540 3 : 15006723301,540 2 : 17506813301,540 1 : 11,5006922981,817 1 : 23,0006922981,817 1 : 34,5006952981,863 1 : 46,0006942981,863 Thinking-1,500 11,877 1,903 54,786 under the 72B model as the thinking-mode training dataset. 3 Data Processing As mentioned above (Section 2), we select GSM8K (Cobbe et al., 2021) and OpenMathRea- soning (Moshkov et al., 2025) as the base datasets for non-thinking and thinking modes, respectively, which are in accordance with the Qwen3 mode design for simple and complex problems. Non-thinking mode dataset. For the non- thinking mode, we use the GSM8K training set. For each experimental setting, we subsample GSM8K into a subset of sizeN NT according to a target ratio ρbetween thinking (N T ) and non-thinking (N NT ) data. We define the ratio as ρ = N T N NT .(1) We sweepρfrom4to1/4(i.e., non-thinking to thinking ratios from4:1to1:4). Since the GSM8K training set contains only7,470examples, to sat- isfy our experimental design for the data ratio, we fix the amount of thinking mode data to1,500ex- amples. With the thinking mode training data fixed at1,500, the corresponding non-thinking training data size ranges from 375 to 6,000. Thinking mode dataset. To enable controlled comparisons across different fusion strategies and data ratios, we fix the number of thinking mode training examples toN T = 1,500throughout our benchmark. The OpenMathReasoning dataset contains3.2million long chain-of-thought solu- tions; from this corpus, we sample1,875examples. We then randomly split these samples into a8:2 Table 2: Statistics of test data for thinking and non- thinking evaluation. Mean denotes the mean length of the reasoning traces, Min denotes the minimum length of the reasoning traces, and Max denotes the maximum length of the reasoning traces. ModeCountMeanMinMax Non-thinking3756962981,863 Thinking37511,9921,84069,268 train/test partition. Detailed statistics of the think- ing and non-thinking training data are presented in Table 1. For uniform evaluation, we sample375test ex- amples from the GSM8K and OpenMathReasoning test sets for the non-thinking and thinking modes, respectively. Detailed statistics of the thinking and non-thinking test data are presented in Table 2. The thinking mode test data have a mean length of11,992, with a maximum length of69,268and a minimum length of1,840. In contrast, the non- thinking mode test data have a mean length of696, with a maximum length of1,863and a minimum length of 298. After data selection, to align with the Qwen3 of- ficial chat template settings, we further process the GSM8K subset into non-thinking mode data and the OpenMathReasoning subset into thinking mode data. For non-thinking mode data, we insert the <think></think>tags before the solutions. For thinking mode data, we wrap the long-form rea- soning solutions with the<think></think>tags. To simplify the evaluation process, we wrap all final answers in the tag. Examples of non-thinking and thinking data are shown in Ap- pendix A.4. 4 Methodology We study Thinking Mode Fusion (TMF) as a post-training problem for large language models (Qwen3 models), where the model support two dis- tinct modes: (i) long-form reasoning for hard prob- lems and (i) concise responses for simple prob- lems. Our goal is to evaluate how TMF is affected by two controllable factors during supervised fine- tuning (SFT): (a) training schedule (the ordering or interleaving of thinking and non-thinking mode training) and (b) data ratio (the relative amount of thinking versus non-thinking data, ranging from1:4 to4:1, while fixing the amount of thinking mode training data). We use the non-thinking and think- ing test sets constructed in Section 3 to evaluate the uniformity of model performance across modes. Figure 2 provides an overview of our experimental design. 4.1 Supervised Fine-Tuning with Two Modes We construct two types of training data (thinking and non-thinking) and feed them into a unified supervised fine-tuning (SFT) framework with dif- ferent training schedules and data ratios. As mentioned in Section 3, we convert ex- amples into the model’s chat template, with the mode specified by the presence or absence of reasoning content within the<think></think> tags. For thinking-mode training, we use the con- structed thinking-mode training data, in which long-form reasoning solutions are wrapped in the <think></think>tags. For non-thinking-mode training, we use the constructed non-thinking-mode training data, which contain short solutions follow- ing the <think></think> tags. After training, we extract final answers from the tag, as described in Section 3. Finally, we evaluate model performance in both thinking and non-thinking modes using the corresponding test sets in Table 2. 4.2 Data Ratio Control As described in Section 3, we construct thinking mode and non-thinking mode training data with a wide range of ratios1:4,1:3,1:2,1:1,2:1,3:1,4:1 by fixing the number of thinking mode training examples. 4.3 Training Schedule Given a fixed thinking setD T and a ratio-specific non-thinking subsetD (ρ) NT , TMF training reduces to constructing a single SFT streamSby a scheduling function S = SCHEDULE D T ,D (ρ) NT ;m ,(2) wheremspecifies how the two modes are arranged during training. We investigate three schedules that cover sequential curricula and interleaving. Sequential:T-NT. We place all thinking mode examples before all non-thinking mode examples: S T-NT = [D T ∥D (ρ) NT ].(3) This schedule tests whether updates from non- thinking data applied after reasoning training dilute or overwrite reasoning behaviors, and whether rea- soning training provides transferable benefits to subsequent concise answering. Thinking Mode Fusion Accuracy Thinking data Non- thinking data Length Data Ratio Training Schedule 1:41:31:21:12:13:14:1 NT-T T-NT Mix Non-Thinking Data Non-Thinking Data Thinking Data Thinking Data NNTT Non-Thinking Mode Thinking Mode Data Ratio Length Figure 2: Overview of Thinking Mode Fusion training design. We construct two datasets: thinking mode and non-thinking mode. We vary the data ratio (T:NT from1:4to4:1) and schedule the resulting SFT stream using three strategies: T-NT (first thinking then non-thinking), NT-T (non-thinking then thinking), and Mix (interleaved to match the target ratio). Each configuration fine-tunes Qwen3 model and is evaluated on both thinking and non-thinking benchmarks to quantify interference and trade-offs. Sequential: NT-T. We reverse the order: S NT-T = [D (ρ) NT ∥D T ].(4) This schedule investigates the opposite direc- tion of interference: whether late-stage reason- ing supervision disrupts concise response patterns learned from non-thinking data and whether non- thinking alignment persists under subsequent long- reasoning updates. Interleaved:Mix. We interleave the two modes of data to the target ratio during training. In partic- ular, we use a simple 1-to-ninterleaving procedure: whenρ ≤ 1(thinking is less frequent), we inter- leave 1 T (thinking mode) followed bynNT (non- thinking mode) examples, withn≈ round(1/ρ); whenρ > 1(thinking is more frequent), we in- terleave 1 NT followed bynT examples, with n≈ round(ρ). This schedule makes the intended TMF usage where the model repeatedly alternates between the two modes, enabling us to test whether frequent mode switching alleviates the degradation observed under purely sequential curricula. 4.4 Full Factorial Design Our TMF analysis is defined by the Cartesian prod- uct of data ratios and schedules: (ρ,m)∈4,3,2,1, 1 2 , 1 3 , 1 4 ×T-NT,NT-T,Mix.(5) For each configuration(ρ,m), we build the corre- sponding SFT trainingSwhile keeping the base model, supervision format, and optimization hyper- parameters fixed. This controlled setup allows us to attribute performance changes to (i) how much thinking supervision is mixed with non-thinking data and (i) how the two modes are scheduled during training, thereby directly quantifying inter- ference and trade-offs induced by TMF. We finally evaluate both the accuracy and the reasoning length of model responses, as reasoning length has been shown to reflect reasoning quality in mathematical problem solving (Sprague et al., 2025; Jin et al., 2024; Tutek et al., 2025). 5 Experiment and Analysis 5.1 Experiment Setup Our study aims to answer four research questions: RQ1: Do the non-thinking and thinking modes affect each other’s performance? Although Qwen3 adopts Thinking Mode Fu- sion to separate long-form and concise reasoning into different fine-tuning modes, training these two modes jointly may still lead to performance trade- offs or interference in the mathematical domain. RQ2: How does the different training sched- ule affect Thinking Mode Fusion? Thinking Mode Fusion is introduced by first fine-tuning on concise reasoning traces and then fine-tuning on long-form reasoning traces (Yang et al., 2025). However, in practice, non-thinking and thinking modes can be trained in different orders. As de- scribed in Section 4.3, we consider three training schedules, T-NT, NT-T, Mix, and evaluate model performance under different training schedules. RQ3: How does the different data ratio affect Thinking Mode Fusion? In real-world applica- tions, short-form reasoning data are far more abun- dant than high-quality long-form reasoning data. Therefore, the data ratio between thinking and non- thinking modes is crucial for understanding the effectiveness of Thinking Mode Fusion. RQ4: What is the quantitative relationship between the non-thinking and thinking modes? After analyzing the effects of training schedules and data ratios, it is essential to quantify the re- lationship between the two modes to better guide TMF training for LLMs. All detailed experimental settings and baselines can be found in Appendix A. 5.2 RQ1: Interaction between Non-Thinking and Thinking Mode The accuracy and reasoning length performance on the test set under training with different data ratios and training schedule settings are presented in Ta- ble 3. Rows correspond to three schedules with two modes: T-NT (thinking before non-thinking), NT-T (non-thinking before thinking), and Mix (in- terleaved training). Columns are organized by the thinking to non-thinking (T:NT) data ratio from 1:4to4:1, reporting Len (average response length) and Acc (accuracy) for three runs. The rightmost Mean column averages results across all data ratios, while the Mean row in each mode block averages over training schedules. The results indicate that the non-thinking and thinking modes interact during Thinking Mode Fusion (TMF) training. When averaging results across all data ratios, the Mix training schedule achieves the highest accuracy on both the think- ing and non-thinking test sets, with accuracies of 0.208and0.695, respectively. In thinking-mode evaluation, the NT-T training schedule also attains the highest accuracy while producing the longest average response length. When averaging results across all training schedules, the thinking mode achieves its highest accuracy at the largest thinking data ratio (T:NT=4:1), whereas the non-thinking mode achieves its highest accuracy at the second highest non-thinking data ratio (T:NT=1:3). Both thinking and non-thinking modes produce the longest response length at the largest think- ing data ratio (T:NT=4:1) when averaging over the BaselineT-NTNT-TMixBaselineT-NTNT-TMix 0.62 0.64 0.66 0.68 0.70 0.72 Non-thinking Non-thinkingThinking BaselineT-NTNT-TMix 0.170 0.175 0.180 0.185 0.190 0.195 0.200 Thinking (a) Accuracy on non-thinking and thinking mode test sets after TMF. BaselineT-NTNT-TMixBaselineT-NTNT-TMix 230 240 250 260 270 280 Non-thinking Non-thinkingThinking BaselineT-NTNT-TMix 5000 5200 5400 5600 5800 6000 6200 6400 Thinking (b) Response length on non-thinking and thinking mode test sets after TMF. Figure 3: The effect of different training schedules under a fixed thinking to non-thinking data ratio of 1:1. training schedules. Similarly, both modes produce the longest response under the NT-T training sched- ule when averaging over the data ratio. These re- sults are consistent with the intuition and prior work that tuning with long-form data at the end tends to yield longer responses (Zhao et al., 2024; Köksal et al., 2024). Overall, these results demonstrate that the per- formance of TMF is affected by the interaction between thinking and non-thinking modes, as mod- ulated by variations in the data ratio and the training schedule. The Mix training schedule achieves the highest overall accuracy, while increasing the think- ing data ratio consistently improves thinking-mode performance and leads to longer responses. 5.3 RQ2: Effect of Training Schedule As shown in Figure 3, when fixing the data ratio at 1:1, the results indicate that the choice of train- ing schedule has a substantial impact on Thinking Mode Fusion training. The performance of the untuned base model on the test set is used as a base- line and is indicated by the gray bar. The orange, blue, and green bars represent the performance un- der the T-NT, NT-T, and Mix training schedules, respectively. In terms of the non-thinking mode, the Mix schedule achieves the highest accuracy, while the Table 3: Performance comparison of accuracy and reasoning length under different data ratios and training schedule settings. We report Len (average response length) and Acc (accuracy) for three runs. The rightmost Mean column averages results across all data ratios, while the Mean row in each mode block averages over training schedules. The Mix training schedule achieves the highest overall accuracy, while increasing the thinking data ratio consistently improves thinking-mode performance and leads to longer responses. Data Ratio (T:NT, T=1500) ModeSetting1:41:31:21:12:13:14:1 Mean LenAccLenAccLenAccLenAccLenAccLenAccLenAccLenAcc T-NT271.3 0.737 245.9 0.692 254.8 0.700 246.8 0.640 316.6 0.695 295.1 0.667 328.8 0.676279.90.687 NT-T245.1 0.664 260.1 0.716 278.2 0.684 270.3 0.692 269.1 0.696 343.6 0.697 399.8 0.693295.20.692 Mix268.3 0.703 254.3 0.706 255.1 0.676 272.1 0.717 273.7 0.698 326.7 0.677 360.5 0.687287.20.695 Non-thinking Mean 261.60.701253.40.705262.70.687263.10.683286.50.696321.80.680363.00.685– T-NT 5455.0 0.188 5632.5 0.198 5065.0 0.174 5519.8 0.186 6242.4 0.220 6858.5 0.224 7027.6 0.2215971.50.202 NT-T 5731.7 0.200 5779.5 0.192 5938.7 0.204 5606.7 0.196 6225.7 0.212 6682.2 0.224 7080.8 0.2276149.30.208 Mix5877.1 0.208 5333.3 0.204 5438.8 0.190 5911.4 0.193 6211.7 0.215 7000.4 0.218 6888.5 0.2256094.50.208 Thinking Mean5687.90.1995581.80.1985480.80.1895679.30.1926226.60.2166847.00.2226999.00.224– T-NT schedule yields the lowest accuracy. More- over, the T-NT schedule shows a degradation in performance, as the base model already achieves an accuracy of0.683without tuning. The response length exhibits the same pattern as accuracy. In terms of thinking mode, the NT-T sched- ule achieves the highest accuracy, while the T-NT schedule results in the lowest accuracy. Compared with the performance of the base model, all train- ing schedules lead to accuracy improvements. The Mix schedule produces the longest response length, which differs from the trend observed in the non- thinking mode. 5.4 RQ3: Effect of Data Ratio The trend comparison of accuracy and reason- ing length under different data ratios and training schedules is presented in Figure 4. The orange, blue, and green lines represent the performance under the T-NT, NT-T, and Mix training schedules, respectively. The dashed lines indicate fitted trend lines. The accuracy of the non-thinking mode shows a decreasing trend as the data ratio increases under the T-NT and NT-T training schedules, whereas it shows an increasing trend as the data ratio increases under the Mix training schedule. However, perfor- mance is sensitive to the choice of training sched- ule under the same data ratio. The T-NT schedule achieves the highest performance when the data ratio is1:4, while the Mix schedule achieves the highest performance when the data ratio is4:1. When non-thinking mode data demonstrate an ad- vantage, the T-NT schedule is preferable; when thinking mode data demonstrate an advantage, the Mix schedule is preferable. When the data ratio is 1:1, the Mix schedule yields the best performance. Moreover, the response length of the non-thinking mode shows a consistent increase as the data ratio increases across all training schedules. These re- sults indicate that different data ratios should be paired with different training schedules. The accuracy of the thinking mode exhibits a consistent increase as the data ratio increases across all training schedules. The Mix schedule achieves the highest accuracy when the data ratio is1:4, while the NT-T schedule achieves the highest accu- racy when the data ratio is4:1. Notably, the think- ing mode training data are fixed at1,500exam- ples, indicating that increasing the amount of non- thinking mode data leads to performance degrada- tion in the thinking mode. The reasoning length of the thinking mode also shows a increase as the data ratio increases across all training schedules. Overall, the data ratio plays a critical role in TMF by systematically trading off non-thinking and thinking performance: higher non-thinking proportions improve concise-answer accuracy but degrade long-form reasoning accuracy. Moreover, the optimal training schedule is strongly coupled with the data ratio. 5.5 RQ4: The Quantitative Relationship between Two Modes We further explore the quantitative relationship be- tween the non-thinking and thinking modes under different training schedules and data ratios, and present a scatter plot in Figure 5. The orange, blue, and green scatter points represent accuracy under the T-NT, NT-T, and Mix training schedules, re- spectively, while the dashed line indicates the fitted trend line of aggregation of the training schedules. 1:41:31:21:12:13:14:1 0.64 0.66 0.68 0.70 0.72 0.74 T-NTNT-TMix (a) Non-thinking accuracy 1:41:31:21:12:13:14:1 250 300 350 400 T-NTNT-TMix (b) Non-thinking length 1:41:31:21:12:13:14:1 0.18 0.20 0.22 T-NTNT-TMix (c) Thinking accuracy 1:41:31:21:12:13:14:1 5000 5500 6000 6500 7000 T-NTNT-TMix (d) Thinking length Figure 4: Performance comparison of accuracy and reasoning length under different data ratios and training schedule settings on non-thinking mode, averaged over three runs. Higher non-thinking ratios improve non-thinking accuracy but degrade thinking accuracy; under the Mix schedule, non-thinking accuracy increases with the data ratio, while thinking accuracy consistently increases across all schedules. 0.6250.6500.6750.7000.7250.750 Non-thinking accuracy 0.16 0.18 0.20 0.22 0.24 Thinking accuracy T-NT NT-T MixAggregated medianOverall trend r = -0.58, R 2 = 0.34, slope = -0.91 Figure 5: Scatter plot of different training schedules and data ratios. The dashed line indicates the fitted trend line of aggregation of the training schedules. The line shows a clear negative correlation (r = −0.58,R 2 = 0.34), with a regression slope of−0.91. The decreasing trend reveals a clear trade-off between the two modes. Specifically, non-thinking and thinking accuracies exhibit a negative corre- lation (r = −0.58), with a coefficient of determi- nation ofR 2 = 0.34. The estimated regression slope (−0.91) indicates that a0.01increase in non- thinking mode performance is associated with an average decrease of approximately0.0091in think- ing mode performance. The two modes hold different and potentially competing objectives: the non-thinking mode fa- vors concise responses, whereas the thinking mode emphasizes long-form reasoning. Consequently, encapsulating these conflicting modes induces a negative correlation. 6 Conclusion In this work, we present a systematic study of the effects of the training schedule and the data ra- tio in TMF, revealing a trade-off between the two modes in which increased non-thinking supervision degrades thinking mode performance. Moreover, we show that the training schedule modulates this trade-off: when averaging results across all data ratios, the Mix training schedule achieves the high- est accuracy on both the thinking and non-thinking test sets. Our results show that the optimal training schedule depends on the data ratio, and that the Mix training schedule performs well across varying data ratios for both modes. Finally, we quantify a nega- tive correlation between non-thinking and thinking mode supervision, highlighting an inherent tension between these two modes. 7 Limitations Our analysis is restricted to mathematical reason- ing, and it remains unclear whether the observed interference patterns generalize to other domains, such as code generation. Second, we fix the amount of thinking mode data to enable controlled com- parisons; exploring regimes in which both think- ing and non-thinking data scale jointly may un- cover richer dynamics. Finally, while we quantify the trade-off between concise answering and long- form reasoning, developing training strategies that explicitly mitigate this tension remains an open challenge. 8 Ethics Statement We do not foresee any particular ethical concerns with our study, which analyzes existing Qwen3 models and is unlikely to lead to unforeseen uses of those models. We use two mathematics datasets, GSM8K and OpenMathReasoning, both of which are publicly available. The toolkits used in our experiments, including Transformers and TRL, are also publicly available under Apache-2.0 License. References Daman Arora and Andrea Zanette. 2025. Training lan- guage models to reason efficiently. In The Thirty- ninth Annual Conference on Neural Information Pro- cessing Systems. Karl Cobbe, Vineet Kosaraju, Mohammad Bavarian, Mark Chen, Heewoo Jun, Lukasz Kaiser, Matthias Plappert, Jerry Tworek, Jacob Hilton, Reiichiro Nakano, Christopher Hesse, and John Schulman. 2021. Training verifiers to solve math word prob- lems. Preprint, arXiv:2110.14168. DeepSeek-AI. 2025. Deepseek-r1: Incentivizing rea- soning capability in llms via reinforcement learning. Preprint, arXiv:2501.12948. Bofei Gao, Feifan Song, Zhe Yang, Zefan Cai, Yibo Miao, Qingxiu Dong, Lei Li, Chenghao Ma, Liang Chen, Runxin Xu, Zhengyang Tang, Benyou Wang, Daoguang Zan, Shanghaoran Quan, Ge Zhang, Lei Sha, Yichang Zhang, Xuancheng Ren, Tianyu Liu, and Baobao Chang. 2025. Omni-MATH: A univer- sal olympiad level mathematic benchmark for large language models. In The Thirteenth International Conference on Learning Representations. Edward J Hu, yelong shen, Phillip Wallis, Zeyuan Allen- Zhu, Yuanzhi Li, Shean Wang, Lu Wang, and Weizhu Chen. 2022. LoRA: Low-rank adaptation of large language models. In International Conference on Learning Representations. Mingyu Jin, Qinkai Yu, Dong Shu, Haiyan Zhao, Wenyue Hua, Yanda Meng, Yongfeng Zhang, and Mengnan Du. 2024. The impact of reasoning step length on large language models. In Findings of the Association for Computational Linguistics: ACL 2024, pages 1830–1842, Bangkok, Thailand. Associ- ation for Computational Linguistics. Marek Kadl ˇ cík, Michal Štefánik, Ondrej Sotolar, and Vlastimil Martinek. 2023. Calc-X and calcformers: Empowering arithmetical chain-of-thought through interaction with symbolic systems. In Proceedings of the 2023 Conference on Empirical Methods in Natural Language Processing, pages 12101–12108, Singapore. Association for Computational Linguis- tics. Team Kimi, Angang Du, Bofei Gao, Bowei Xing, Changjiu Jiang, Cheng Chen, Cheng Li, Chenjun Xiao, Chenzhuang Du, Chonghua Liao, and 1 others. 2025. Kimi k1. 5: Scaling reinforcement learning with llms. arXiv preprint arXiv:2501.12599. Takeshi Kojima, Shixiang (Shane) Gu, Machel Reid, Yu- taka Matsuo, and Yusuke Iwasawa. 2022. Large lan- guage models are zero-shot reasoners. In Advances in Neural Information Processing Systems, volume 35, pages 22199–22213. Curran Associates, Inc. Abdullatif Köksal, Timo Schick, Anna Korhonen, and Hinrich Schuetze. 2024. LongForm: Effective in- struction tuning with reverse instructions. In Find- ings of the Association for Computational Linguistics: EMNLP 2024, pages 7056–7078, Miami, Florida, USA. Association for Computational Linguistics. Jia LI, Edward Beeching, Lewis Tunstall, Ben Lipkin, Roman Soletskyi, Shengyi Costa Huang, Kashif Rasul, Longhui Yu, Albert Jiang, Ziju Shen, Zihan Qin, Bin Dong, Li Zhou, Yann Fleureau, Guillaume Lample, and Stanislas Polu. 2024. Numinamath.[https://huggingface.co/ AI-MO/NuminaMath-CoT](https://github.com/ project-numina/aimo-progress-prize/blob/ main/report/numina_dataset.pdf). Yuan Li, Zhengzhong Liu, and Eric P. Xing. 2025. Data mixing optimization for supervised fine-tuning of large language models. In Forty-second International Conference on Machine Learning. Qian Liu, Xiaosen Zheng, Niklas Muennighoff, Guang- tao Zeng, Longxu Dou, Tianyu Pang, Jing Jiang, and Min Lin. 2025. Regmix: Data mixture as regression for language model pre-training. In The Thirteenth International Conference on Learning Representa- tions. Ivan Moshkov, Darragh Hanley, Ivan Sorokin, Shub- ham Toshniwal, Christof Henkel, Benedikt Schif- ferer, Wei Du, and Igor Gitman. 2025. Aimo-2 win- ning solution: Building state-of-the-art mathematical reasoning models with openmathreasoning dataset. Preprint, arXiv:2504.16891. OpenAI. 2024. Openai o1 system card. Preprint, arXiv:2412.16720. OpenAI. 2025. Gpt-5.1: A smarter, more conversational chatgpt. Arkil Patel, Satwik Bhattamishra, and Navin Goyal. 2021. Are NLP models really able to solve simple math word problems? In Proceedings of the 2021 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, pages 2080–2094, Online. Association for Computational Linguistics. Yi Shen, Jian Zhang, Jieyun Huang, Shuming Shi, Wen- jing Zhang, Jiangze Yan, Ning Wang, Kai Wang, Zhaoxiang Liu, and Shiguo Lian. 2025. DAST: Difficulty-adaptive slow-thinking for large reasoning models. In Proceedings of the 2025 Conference on Empirical Methods in Natural Language Processing: Industry Track, pages 2322–2331, Suzhou (China). Association for Computational Linguistics. Zayne Rea Sprague, Fangcong Yin, Juan Diego Ro- driguez, Dongwei Jiang, Manya Wadhwa, Prasann Singhal, Xinyu Zhao, Xi Ye, Kyle Mahowald, and Greg Durrett. 2025. To cot or not to cot? chain-of- thought helps mainly on math and symbolic reason- ing. In The Thirteenth International Conference on Learning Representations. Gemma Team. 2025.Gemma 3 technical report. Preprint, arXiv:2503.19786. Shubham Toshniwal, Wei Du, Ivan Moshkov, Branislav Kisacanin, Alexan Ayrapetyan, and Igor Gitman. 2025. Openmathinstruct-2: Accelerating AI for math with massive open-source instruction data. In The Thirteenth International Conference on Learning Representations. Shubham Toshniwal, Ivan Moshkov, Sean Narenthi- ran, Daria Gitman, Fei Jia, and Igor Gitman. 2024. Openmathinstruct-1: A 1.8 million math instruction tuning dataset. In Advances in Neural Information Processing Systems, volume 37, pages 34737–34774. Curran Associates, Inc. Martin Tutek, Fateme Hashemi Chaleshtori, Ana Marasovic, and Yonatan Belinkov. 2025. Measuring chain of thought faithfulness by unlearning reasoning steps. In Proceedings of the 2025 Conference on Empirical Methods in Natural Language Processing, pages 9946–9971, Suzhou, China. Association for Computational Linguistics. Shouren Wang, Wang Yang, Xianxuan Long, Qi- fan Wang, Vipin Chaudhary, and Xiaotian Han. 2025.Demystifying hybrid thinking: Can llms truly switch between think and no-think? Preprint, arXiv:2510.12680. Xuezhi Wang, Jason Wei, Dale Schuurmans, Quoc V Le, Ed H. Chi, Sharan Narang, Aakanksha Chowdhery, and Denny Zhou. 2023. Self-consistency improves chain of thought reasoning in language models. In The Eleventh International Conference on Learning Representations. Jason Wei, Xuezhi Wang, Dale Schuurmans, Maarten Bosma, brian ichter, Fei Xia, Ed Chi, Quoc V Le, and Denny Zhou. 2022. Chain-of-thought prompt- ing elicits reasoning in large language models. In Advances in Neural Information Processing Systems, volume 35, pages 24824–24837. Curran Associates, Inc. Han Wu, Yuxuan Yao, Shuqi Liu, Zehua Liu, Xiaojin Fu, Xiongwei Han, Xing Li, Hui-Ling Zhen, Tao Zhong, and Mingxuan Yuan. 2025. Unlocking efficient long- to-short llm reasoning with model merging. arXiv preprint arXiv:2503.20641. An Yang, Anfeng Li, Baosong Yang, Beichen Zhang, Binyuan Hui, Bo Zheng, Bowen Yu, Chang Gao, Chengen Huang, Chenxu Lv, Chujie Zheng, Day- iheng Liu, Fan Zhou, Fei Huang, Feng Hu, Hao Ge, Haoran Wei, Huan Lin, Jialong Tang, and 41 others. 2025. Qwen3 technical report. Preprint, arXiv:2505.09388. Jiasheng Ye, Peiju Liu, Tianxiang Sun, Jun Zhan, Yun- hua Zhou, and Xipeng Qiu. 2025a. Data mixing laws: Optimizing data mixtures by predicting lan- guage modeling performance. In The Thirteenth In- ternational Conference on Learning Representations. Yixin Ye, Zhen Huang, Yang Xiao, Ethan Chern, Shijie Xia, and Pengfei Liu. 2025b. LIMO: Less is more for reasoning. In Second Conference on Language Modeling. Liang Zeng, Liangjun Zhong, Liang Zhao, Tianwen Wei, Liu Yang, Jujie He, Cheng Cheng, Rui Hu, Yang Liu, Shuicheng Yan, Han Fang, and Yahui Zhou. 2024. Skywork-math: Data scaling laws for mathematical reasoning in large language models – the story goes on. Preprint, arXiv:2407.08348. Jiajie Zhang, Nianyi Lin, Lei Hou, Ling Feng, and Juanzi Li. 2025. AdaptThink: Reasoning models can learn when to think. In Proceedings of the 2025 Conference on Empirical Methods in Natural Lan- guage Processing, pages 3716–3730, Suzhou, China. Association for Computational Linguistics. Hao Zhao, Maksym Andriushchenko, Francesco Croce, and Nicolas Flammarion. 2024. Long is more for alignment: a simple but tough-to-beat baseline for in- struction fine-tuning. In Proceedings of the 41st Inter- national Conference on Machine Learning, ICML’24. JMLR.org. A Experimental Settings Baseline.To study whether the non-thinking and thinking modes affect each other’s performance, we adopt a TMF training setting with a data ratio of T:NT = 1:1 and a training schedule of NT-T as the baseline. To analyze the effect of training schedules on Thinking Mode Fusion, we use the performance of the base model without any fine- tuning as the baseline. To study the effect of data ratios on Thinking Mode Fusion, we again adopt TMF training with a data ratio of T:NT = 1:1 and a training schedule of T-NT as the baseline. Overview of Training settings. All experi- ments use continual supervised fine-tuning (SFT) on Qwen3-4B with Transformers and TRL. 23 To isolate TMF effects while maintaining effi- ciency, we apply LoRA (Hu et al., 2022) to the main projection modules (q_proj, k_proj, v_proj, o_proj, gate_proj, up_proj, down_proj) withr=16,α=64, dropout0.0, and no bias. Opti- mization uses AdamW with a cosine learning rate schedule, a learning rate of2× 10 −5 , weight decay 0.001, and 5 warmup steps. All runs use a maxi- mum sequence length of5,120tokens and train for one epoch; results are averaged over three runs. A.1 Base Model and Training Framework We conduct all experiments starting from Qwen3- 4B, an open-weight model that natively supports a unified chat template with optional<think>blocks. All fine-tuning runs are implemented using Trans- formers and TRL with SFTTrainer. To isolate the effects of Thinking Mode Fusion (TMF) strategies, we perform continual supervised fine-tuning (SFT) with a lightweight parameter-efficient setup via LoRA (Hu et al., 2022). Specifically, we apply LoRA to all major projec- tion modules in the transformer blocks:q_proj, k_proj, v_proj, o_proj, gate_proj, up_proj, down_proj . Unless otherwise stated, LoRA uses rankr=16,α=64, dropout0.0, and no bias adaptation. Training is performed in bfloat16 withdevice_map=auto.We enable TF32 for CUDA matmul and cuDNN when available to im- prove throughput. A.2 Training Data To instantiate the two modes in TMF, we use one dataset for long-form reasoning supervision (think- ing mode) and one dataset for short-solution super- vision (non-thinking mode). Thinking mode data. We use OpenMathRea- soning, splitcot) as our thinking mode dataset. Each sample includes a mathproblemand a long 2 https://huggingface.co/docs/transformers 3 https://huggingface.co/docs/trl chain-of-thought stylegenerated_solution. We randomly split the dataset into 80% training and 20% test using a fixed seed. Non-thinking mode data.We use GSM8K, con- figurationmain) as our non-thinking dataset. Each sample contains aquestionand ananswerwith a canonical “####” delimiter. We use the official train/test split. Training subset size. To ensure controlled com- parisons across different fusion ratios, we cap the thinking mode training set to a fixed sizeN T = standard_index(default 1500). For each run, we sample the non-thinking set sizeN NT accord- ing to a target ratioρ = N T /N NT (denoted as reasoning_chat_percentage in code), i.e., N NT = N T · 1 ρ .(6) Non-thinking samples are randomly sub-sampled with a fixed random seed for reproducibility. A.3 Example of Data Thinking mode data: Suppose that ( x, y in mathbbR ) satisfy ( |x-1| leq 2 ) and ( |2y+1| leq 5 ). Let ( M ) and ( m ) be the maximum and minimum of ( xy - x + 3y + 3 ), respectively. Find the value of ( M + m ). Non-thinking mode data: Natalia sold clips to 48 of her friends in April, and then she sold half as many clips in May. How many clips did Natalia sell altogether in April and May? A.4 Prompting and Mode Formatting We unify both datasets under a chat-style supervi- sion format using the model’s native chat template. For thinking mode data, the user instruction explic- itly requests step-by-step reasoning and enforces the final answer format: Please reason step by step, and put your final answer within . Problem: Suppose that ( x, y in mathbbR ) satisfy ( |x-1| leq 2 ) and ( |2y+1| leq 5 ). Let ( M ) and ( m ) be the maximum and minimum of ( xy - x + 3y + 3 ), respectively. Find the value of ( M + m ). <think>Okay, so I need to find the maximum and minimum values of the expression xy - x + 3y + 3 given the constraints |x - 1|≤ 2 and |2y + 1|≤ 5. Then, once I have those, I need to add them together to get M + m. Let me try to break this down step by step. First, let’s understand the constraints. The first one is |x - 1|≤ 2. This means that x is within 2 units of 1. So, solving that inequality, we subtract 1 and take the absolute value into account: |x - 1|≤ 2 implies -2≤ x - 1≤ 2. ... </think> Given the constraints...Therefore, the final answer is:boxed-6. For non-thinking mode data, the instruction re- quests a direct solution with the same final answer constraint: Solve the problem, and put your final answer within . Problem: Natalia sold clips to 48 of her friends in April, and then she sold half as many clips in May. How many clips did Natalia sell altogether in April and May? <think></think> Natalia sold 48/2 = «48/2=24»24 clips in May.Natalia sold 48+24 = «48+24=72»72 clips altogether in April and May.**Final Answer** boxed72. To align with the TMF interface, we wrap non- thinking solutions with an<think>block that is intentionally left empty, followed by a short solu- tion and a boxed final answer. This design enforces a consistent output schema across modes while discouraging long reasoning in the non-thinking mode. A.5 TMF Training Strategies: Order and Mixing We study three training strategies that control the interaction between thinking and non-thinking su- pervision: • T-NT: sequential training where thinking mode samples are placed before non-thinking samples in the SFT stream. •NT-T: sequential training where non-thinking samples are placed before thinking samples. •Mix: an interleaving strategy that alternates the two modes according to the target ra- tioρ.Whenρ ≤ 1(i.e., thinking is less frequent), we interleave1 thinking : n non-thinkingwithn ≈ 1/ρ. When ρ > 1, we interleaven thinking : 1 non-thinking with n≈ ρ. All strategies share the same total number of training examples determined by(N T ,N NT ) un- der each ratio. Solve the problem, and put your final answer within boxed\\. Problem: Natalia sold clips to 48 of her friends in April, and then she sold half as many clips in May. How many clips did Natalia sell altogether in April and May? <think></think> Natalia sold 48/2 = <<48/2=24>>24 clips in May.Natalia sold 48+24 = <<48+24=72>>72 clips altogether in April and May.**Final Answer** boxed\72\. (a) Non-thinking. Please reason step by step, and put your final answer within boxed\\. Problem: Suppose that ( x, y in mathbbR ) satisfy ( |x-1| leq 2 ) and ( |2y+1| leq 5 ). Let ( M ) and ( m ) be the maximum and minimum of ( xy - x + 3y + 3 ), respectively. Find the value of ( M + m ). <think>Okay, so I need to find the maximum and minimum values of the expression xy - x + 3y + 3 given the constraints |x - 1| ≤ 2 and |2y + 1| ≤ 5. Then, once I have those, I need to add them together to get M + m. Let me try to break this down step by step. First, let's understand the constraints. The first one is |x - 1| ≤ 2. This means that x is within 2 units of 1. So, solving that inequality, we subtract 1 and take the absolute value into account: |x - 1| ≤ 2 implies -2 ≤ x - 1 ≤ 2. ... </think> Given the constraints...Therefore, the final answer is:boxed-6. (b) Thinking. Figure 6: An overview of the experimental design for analyzing Thinking Mode Fusion under different train- ing orders and data ratios. A.6 Optimization and Hyperparameters We fine-tune the model using AdamW with cosine learning-rate scheduling. Unless otherwise stated, hyperparameters are fixed across all runs: • Learning rate: 2× 10 −5 • Weight decay: 0.001 • Warmup steps: 5 • Epochs: 1 • Per-device batch size: 1 •Gradient accumulation: 6 (effective batch size 6) • Maximum sequence length: 5120 tokens We enable gradient checkpointing during train- ing and disable key-value caching to reduce mem- ory usage. A.7 Evaluation Protocol We evaluate on both tasks to quantify interference and transfer between modes: GSM8K test for non- thinking capability and OpenMathReasoning-mini test for thinking capability. For efficiency and con- trolled comparisons, we sub-sample each test set to⌊0.25· standard_index⌋ examples. Generation settings. We use greedy decod- ing (do_sample=False) with specific maximum generation budgets:max_new_tokens=1024for GSM8K andmax_new_tokens=5120for Open- MathReasoning. The final answer is extracted via pattern matching on the last occurrence of .... Throughput-aware batching and long-tail con- trol.To stabilize evaluation cost across different prompt lengths, we estimate prompt token statistics on a random sample (default 1024 prompts) and set the evaluation prefill truncation threshold to the 95th percentile token length (capped at 8192). In- puts longer than the percentile threshold are evalu- ated with batch size 1 to avoid padding inefficiency; extremely long prompts beyond the hard cap are filtered by default. We additionally report decoding throughput in tokens/second. Metric.We report exact-match accuracy between the extracted boxed prediction and the gold answer. All evaluations are conducted at the end of each epoch via a custom callback and logged to Weights & Biases, together with token statistics and through- put. A.8 Run Management All experiments are executed on a single-node, single-GPU (NVIDIA H100) cluster setting using SLURM array jobs. We sweep over fusion strate- gies and data ratios, and repeat each configuration multiple times with separate run identifiers. Each run logs configurations and evaluation metrics to Weights & Biases for aggregation and analysis.