Efficient Multilingual Reasoning Transfer via Progressive Code-Switching

# Efficient Multilingual Reasoning Transfer via Progressive Code-Switching
Source: [https://arxiv.org/html/2607.00485](https://arxiv.org/html/2607.00485)
### Reward Modeling

PCS then conducts RL training based on the cold\-start model\. Different from conventional response\-level language\-consistency reward by judging the language of the entire response, we compute theSLC​\(T,L\)\\mathrm\{SLC\}\(T,L\)and compare it against a threshold to obtain the language reward\. This reward reflects whether the model’s current code\-switched reasoning behavior matches the desired level specified by our curriculum\. Given the thresholdτ\\tauofSLC​\(T,L\)\\mathrm\{SLC\}\(T,L\)at the current optimization step, we first define our reward system:

- •Repetition penalty \(rrep\\text\{r\}\_\{\\text\{rep\}\}\):We detect repetition in the response to force better reasoning quality\.rrep=1\\text\{r\}\_\{\\text\{rep\}\}=1if no repetition, otherwise 0\. See Appendix[A](https://arxiv.org/html/2607.00485#A1.SSx3)for more details\.
- •Format reward \(rfmt\\text\{r\}\_\{\\text\{fmt\}\}\):rfmt=1\\text\{r\}\_\{\\text\{fmt\}\}=1if the output follows the<think\>\.\.\.</think\>format, otherwise 0\.
- •Accuracy reward \(racc\\text\{r\}\_\{\\text\{acc\}\}\):racc=1\\text\{r\}\_\{\\text\{acc\}\}=1if the answer is correct, otherwise 0\.
- •Step\-level language consistency reward \(rSLC\\text\{r\}\_\{\\text\{SLC\}\}\):We apply regular expressions to remove mathematical content\. Then uselangdetect111https://github\.com/Mimino666/langdetectto identify the language of each step\.rSLC=1\\text\{r\}\_\{\\text\{SLC\}\}=1ifSLC​\(T,L\)≥τ\\mathrm\{SLC\}\(T,L\)\\geq\\tau, otherwise 0\.

As demonstrated in Figure[2](https://arxiv.org/html/2607.00485#Sx2.F2), we first apply a hard gateC=\(rfmt=1∧rrep=1\)C=\(r\_\{\\mathrm\{fmt\}\}=1\\land r\_\{\\mathrm\{rep\}\}=1\); ifCCis not satisfied, the final reward is set to zero\. WhenCCholds, we assignrfinal=1r\_\{\\mathrm\{final\}\}=1to correct responses andrfinal=0\.1r\_\{\\mathrm\{final\}\}=0\.1otherwise\. For correct responses, we additionally apply a step\-level language consistency bonus:rfinal=1\.1r\_\{\\mathrm\{final\}\}=1\.1ifrSLC=1r\_\{\\mathrm\{SLC\}\}=1, andrfinal=1r\_\{\\mathrm\{final\}\}=1ifrSLC=0r\_\{\\mathrm\{SLC\}\}=0\.

rfinal=\{1\.1,if​C∧\(racc=1\)∧\(rSLC=1\),1,if​C∧\(racc=1\)∧\(rSLC=0\),0\.1,if​C∧\(racc=0\),0,otherwise,r\_\{\\text\{final\}\}=\\begin\{cases\}1\.1,&\\text\{if \}C\\land\(r\_\{\\text\{acc\}\}=1\)\\land\(r\_\{\\text\{SLC\}\}=1\),\\\\ 1,&\\text\{if \}C\\land\(r\_\{\\text\{acc\}\}=1\)\\land\(r\_\{\\text\{SLC\}\}=0\),\\\\ 0\.1,&\\text\{if \}C\\land\(r\_\{\\text\{acc\}\}=0\),\\\\ 0,&\\text\{otherwise\},\\end\{cases\}\(2\)C=\(rfmt=1∧rrep=1\)\.C=\(r\_\{\\text\{fmt\}\}=1\\land r\_\{\\text\{rep\}\}=1\)\.\(3\)
This reward design not only enforces a high\-quality reasoning process but also encourages the model to gradually shift its reasoning language while preserving task performance\.

### Progressive Code\-Switching RL

Although our initial model acquires preliminary code\-switched reasoning ability through cold\-start training, its code\-switching ratio \(i\.e\.,SLC​\(T,L\)\\mathrm\{SLC\}\(T,L\)\) remains low and far from fully target\-language reasoning\. We then employ reinforcement learning with curriculum learning to increaseSLC​\(T,L\)\\mathrm\{SLC\}\(T,L\)\.

Based on our reward modeling, we setτ\\tauat 10% from the beginning and progressively increaseτ\\tauto 95% during RL training\. We track a batch\-level curriculum progress metric, denoted asPass​@​SLC​\(τ\)\\mathrm\{Pass@SLC\}\(\\tau\)\. It is defined as the ratio of responses achievingSLC​\(T,L\)≥τ\\mathrm\{SLC\}\(T,L\)\\geq\\tauto the total number of responses that satisfy the format and correctness constraints\. Given an adjustment intervalkkand a marginΔ​τ\\Delta\\tau, we evaluatePass​@​SLC​\(τ\)\\mathrm\{Pass@SLC\}\(\\tau\)everykkRL optimization step and update the threshold by

τ←\{min⁡\(τ\+Δ​τ,0\.95\),if​Pass​@​SLC​\(τ\)≥0\.9,τ,otherwise\.\\tau\\leftarrow\\begin\{cases\}\\min\(\\tau\+\\Delta\\tau,\\;0\.95\),&\\text\{if \}\\mathrm\{Pass@SLC\}\(\\tau\)\\\\ &\\geq 0\.9,\\\\ \\tau,&\\text\{otherwise\}\.\\end\{cases\}\(4\)We setk=40k=40andΔ​τ=0\.1\\Delta\\tau=0\.1in our experiments\. WhenPass​@​SLC​\(τ\)\\mathrm\{Pass@SLC\}\(\\tau\)reaches 0\.9, it suggests that the language\-shifting reward becomes saturated \(and thus less informative\) in the current stage\. Increasingτ\\taumakes the language\-consistency reward informative again and corresponds to a harder curriculum, since a higherτ\\taurequires a larger proportion of reasoning steps in the target language\.

We design a progressively increasingτ\\tauto encourage the model to gradually shift its reasoning language for each step, rather than directly assigning a language reward based onSLC​\(T,L\)\\mathrm\{SLC\}\(T,L\)\(e\.g\., giving a higher reward for a higherSLC​\(T,L\)\\mathrm\{SLC\}\(T,L\)\)\. This is motivated by the fact that transferring reasoning capability from English to the target language via code\-switched reasoning is typically a slow adaptation process\. An overly aggressive strategy, or one lacking an easy\-to\-hard curriculum, may cause the model to over\-optimize language consistency at the expense of the more important goals of cross\-lingual transfer and reasoning performance\.

AsSLC​\(T,L\)\\mathrm\{SLC\}\(T,L\)increases, the policy distribution deviates more substantially from that of the SFT model, which in turn leads to larger KL divergence and unstable training\. In our experiments, an initial KL constraint limits language\-switching efficiency, so we mitigate this issue by resetting the reference model to the current policy whenever the KL loss exceeds an empirical threshold of 0\.2, and then applying a KL penalty with a coefficient of 0\.001\. We apply GRPO to optimize the policy model in our experiments\.

## Experiments

### Experiment Setup

\\rowcolorgray\!26FRPTJAKOTHALL\-AVGEN\\rowcolorwhite Methodsslc&accaccslcslc&accaccslcslc&accaccslcslc&accaccslcslc&accaccslcslc&accacc\\rowcolorgray\!10 MMATHQwen3\-8B0\.027\.08\.10\.025\.06\.70\.123\.16\.30\.022\.45\.90\.022\.76\.40\.024\.2\\rowcolorwhite Prompt Control0\.030\.110\.20\.228\.19\.40\.022\.07\.90\.023\.68\.92\.721\.817\.80\.625\.8\\rowcolorwhite Prefix Control22\.930\.687\.126\.730\.294\.319\.219\.898\.421\.222\.495\.920\.921\.098\.922\.224\.1\\rowcolorwhite SFT14\.132\.283\.112\.834\.370\.59\.89\.899\.020\.823\.392\.525\.325\.499\.416\.625\.9\\rowcolorwhite Naive\-RL0\.141\.66\.50\.239\.59\.00\.140\.33\.70\.238\.810\.10\.139\.97\.10\.146\.1\\rowcolorwhite RLC\-RL30\.935\.096\.532\.935\.197\.030\.930\.999\.726\.828\.198\.229\.529\.899\.830\.239\.8\\rowcolorwhite M\-Thinker33\.241\.693\.936\.441\.293\.735\.136\.695\.731\.937\.792\.733\.136\.892\.134\.045\.4\\rowcolorgray\!8 PCS41\.242\.698\.037\.239\.297\.136\.838\.198\.333\.633\.898\.936\.737\.695\.637\.145\.0\\rowcolorgray\!10MMLU\-ProXmath\\text\{MMLU\-ProX\}\_\{\\text\{math\}\}Qwen3\-8B0\.151\.720\.60\.042\.419\.60\.044\.119\.20\.138\.319\.80\.050\.921\.80\.057\.0\\rowcolorwhite Prompt Control0\.068\.024\.70\.267\.724\.30\.065\.125\.00\.267\.524\.31\.358\.627\.10\.455\.3\\rowcolorwhite Prefix Control39\.548\.490\.440\.745\.894\.328\.929\.198\.523\.324\.098\.139\.539\.899\.034\.455\.8\\rowcolorwhite SFT41\.565\.690\.530\.366\.380\.40\.40\.499\.08\.610\.095\.954\.554\.799\.427\.165\.9\\rowcolorwhite Naive\-RL1\.782\.616\.61\.382\.922\.30\.780\.314\.12\.079\.930\.22\.080\.420\.51\.585\.7\\rowcolorwhite RLC\-RL75\.280\.597\.574\.580\.696\.374\.975\.099\.770\.874\.197\.875\.475\.699\.874\.283\.2\\rowcolorwhite M\-Thinker58\.479\.692\.653\.780\.391\.374\.676\.297\.669\.878\.394\.774\.077\.796\.766\.183\.4\\rowcolorgray\!8 PCS79\.983\.198\.078\.381\.297\.476\.877\.299\.475\.577\.498\.475\.275\.999\.577\.185\.4

Table 2:Main results on MMATH andMMLU\-ProXmath\\text\{MMLU\-ProX\}\_\{\\text\{math\}\}for Qwen3\-8B\-Base model\.#### Backbones and Languages

We adopt Qwen3\-4B\-Base and Qwen3\-8B\-Base as the backbone models\. We evaluate PCS on a diverse set of languages—French \(fr\), Portuguese \(pt\), Japanese \(ja\), Korean \(ko\), and Thai \(th\)—to examine its effectiveness across typologically different languages\.

#### Data

We design consistent settings for different baselines and PCS\. Please refer to Appendix[A](https://arxiv.org/html/2607.00485#A1)\.

#### Evaluation Details

We evaluate on the math subset of MMLU\-ProX\(Xuanet al\.[2025](https://arxiv.org/html/2607.00485#bib.bib24)\)and MMATH\(Luoet al\.[2025](https://arxiv.org/html/2607.00485#bib.bib7)\)\. We use three metrics:Step\-Level Language Consistency \(SLC\), which measures the proportion of reasoning steps in the target language;Accuracy \(Acc\), which measures answer correctness; andSLC&Acc, the percentage of responses that are both correct and language\-consistent \(SLC⩾0\.9\\geqslant 0\.9\), which serves as our primary metric\. We useSLC⩾0\.9\\geqslant 0\.9to tolerate a small amount of language\-agnostic or stray non\-target tokens while still reflecting practically target\-language reasoning\. We report avg@4 on MMATH, which is harder and more stochastic under long reasoning, and avg@1 onMMLU\-ProXmath\\text\{MMLU\-ProX\}\_\{\\text\{math\}\}following the standard zero\-shot setting\. For MMATH, we macro\-average its four subsets \(MATH500, CNMO, AIME2024, and AIME2025\) to account for their varying difficulty levels\.

#### Baselines

- •Qwen3\-4/8B:We evaluate the post\-trained models in thinking mode under the same context limit\.
- •Prompt Control:Wanget al\.\([2025a](https://arxiv.org/html/2607.00485#bib.bib9)\)append language\-control instructions at inference time \(see Figure[8](https://arxiv.org/html/2607.00485#A3.F8)\)\.
- •Prefix Control:FollowingQiet al\.\([2025a](https://arxiv.org/html/2607.00485#bib.bib20)\), we add “Okay” in the question language after “<think\>” to steer the reasoning language without parameter updates\.
- •SFT:Fine\-tunes the base model on language\-consistent supervised data distilled by DeepSeek\-V3\.2\-Exp\.
- •Naive RL:Optimizes the SFT model only by response correctness\.
- •SoftLC RL:\(Mistral\-AIet al\.[2025](https://arxiv.org/html/2607.00485#bib.bib8)\)adds a soft response\-level language reward \(0\.1\) to Naive RL for target\-language responses\.
- •M\-Thinker:Zhanget al\.\([2026](https://arxiv.org/html/2607.00485#bib.bib11)\)adds cross\-lingual thinking alignment rewards using DeepSeek\-V3\. We reproduce M\-Thinker under our experimental settings\.

### Experiment Results

Tables[Cold\-Start](https://arxiv.org/html/2607.00485#Sx2.SSx2)and[Experiment Setup](https://arxiv.org/html/2607.00485#Sx3.SSx1)present the main results on MMATH andMMLU\-ProXmath\\text\{MMLU\-ProX\}\_\{\\text\{math\}\}\. Across both benchmarks and all five target languages \(fr/pt/ja/ko/th\), PCS consistently achieves the best overall performance under our primary metric,SLC&Acc\.

#### PCS achieves the best balance between correctness and target\-language reasoning\.

Compared with the post\-trained model and inference\-time control methods, PCS substantially improvesSLCwhile maintaining strongAcc\. Prompt Control provides only weak steering over the reasoning language, whereas Prefix Control enforces target\-language reasoning more aggressively but often hurts accuracy\. This suggests that directly constraining the reasoning language at inference time is insufficient for effective transfer of multilingual reasoning\. In contrast, PCS improves language consistency through training\-time transfer, leading to much stronger joint performance\.

#### PCS outperforms both SFT and RL baselines\.

SFT on distilled multilingual traces improves language consistency, but the quality and scale of supervised multilingual reasoning data limit its gains\. We also observe that SFT can introduce severe repetition issues in some languages; for example, in Table[Cold\-Start](https://arxiv.org/html/2607.00485#Sx2.SSx2), Japanese exhibits a substantial performance drop caused by repetition\. Naive RL preserves relatively strongAccbut typically falls back to English reasoning, resulting in very lowSLC\. SoftLC RL improvesSLCwith response\-level language rewards, yet remains clearly behind PCS onSLC&Acc\. Overall, these results show that neither SFT nor RL with a language consistency reward is sufficient: effective multilingual reasoning transfer requires a gradual shift of the reasoning language\.

#### PCS enables stable language transfer with less supervision\.

Across all benchmarks, PCS achieves the highestSLC, reaching roughly 96–98% on average while preserving competitive accuracy\. Compared with M\-Thinker, PCS attains comparable overall performance without relying on distilled target\-language reasoning traces from a stronger LRM or supervision from a stronger judge model\. In our reproduction, M\-Thinker is more vulnerable to reward\-hacking under response\-level language control, leading to lowerSLCdespite relatively strong task performance\. In addition, M\-Thinker requires online reward evaluation by an external judge model, which introduces non\-trivial latency into RL training and can reduce overall hardware utilization\. While such overhead may be alleviated through a larger judge\-serving setup, this comes at the cost of additional computing resources\. By contrast, PCS only uses lightweight language detection, yielding a more resource\-efficient transfer process\. We analyze these behaviors in more detail in the following sections\.

## Analysis

### Mitigating Reward Hacking of Language Consistency Reward

![Refer to caption](https://arxiv.org/html/2607.00485v1/x3.png)Figure 3:Part of M\-Thinker’s thinking process\.The SoftLC RL and M\-Thinker baselines both use response\-level language consistency rewards, which reduces language identification to a binary decision based on off\-the\-shelf language detection tools\. In our experiments, this design is largely effective for SoftLC RL\. However, the same response\-level constraint becomes ineffective for M\-Thinker\. As shown in Figure[3](https://arxiv.org/html/2607.00485#Sx4.F3), when combined with a cross\-lingual thinking alignment \(CTA\) reward provided by a stronger judge model, M\-Thinker often produces low\-quality “target\-language” reasoning that frequently contains Chinese and English words\. This suggests that the policy learns tohackthe response\-level language constraint: it exploits stronger English/Chinese reasoning patterns to maximize the CTA reward while inserting enough target\-language cues to pass a coarse language detector, rather than genuinely target\-language reasoning\. We provide a detailed case study of this reward\-hacking issue in Appendix[E](https://arxiv.org/html/2607.00485#A5)\.

### Reasoning\-Language Transfer Dynamics

![Refer to caption](https://arxiv.org/html/2607.00485v1/x4.png)Figure 4:Training curves forSLC​\(T,L\)\\mathrm\{SLC\}\(T,L\),τ\\tau, and Acc of PCS\.Figure[4](https://arxiv.org/html/2607.00485#Sx4.F4)shows the evolution of accuracy,τ\\tau, andSLC​\(T,L\)\\mathrm\{SLC\}\(T,L\)during training\. Asτ\\tauincreases,SLC​\(T,L\)\\mathrm\{SLC\}\(T,L\)steadily rises and eventually approaches 100%, indicating nearly full target\-language reasoning\. However, increasingSLC​\(T,L\)\\mathrm\{SLC\}\(T,L\)also makes optimization harder, since strongerSLC​\(T,L\)\\mathrm\{SLC\}\(T,L\)constraints can conflict with task performance: pushing the model toward fewer English reasoning steps often tends to reduce accuracy\. To avoid this trade\-off, we assign a higher priority to correctness in the reward design\. As a result, accuracy remains relatively stable whileSLC​\(T,L\)\\mathrm\{SLC\}\(T,L\)improves substantially\. Overall, PCS successfully shifts the reasoning language with only a small performance loss\.

### The Importance of Curriculum

We introduce a curriculum\-learning paradigm into PCS based onSLC​\(T,L\)\\mathrm\{SLC\}\(T,L\)\. To investigate whether this curriculum schedule is necessary, we design an ablation study in which the curriculum is removed and replaced with a reward\-shaping strategy that directly and continuously incentivizes higherSLC​\(T,L\)\\mathrm\{SLC\}\(T,L\)values\.

rfinal=\{1\+0\.1⋅SLC​\(T,L\),if​C∧\(racc=1\),0\.1,if​C∧\(racc=0\),0,otherwise,r\_\{\\text\{final\}\}=\\begin\{cases\}1\+0\.1\\cdot\\mathrm\{SLC\}\(T,L\),&\\text\{if \}C\\land\(r\_\{\\text\{acc\}\}=1\),\\\\ 0\.1,&\\text\{if \}C\\land\(r\_\{\\text\{acc\}\}=0\),\\\\ 0,&\\text\{otherwise\},\\end\{cases\}\(5\)C=\(rfmt=1∧rrep=1\)\.C=\(r\_\{\\text\{fmt\}\}=1\\land r\_\{\\text\{rep\}\}=1\)\.\(6\)
We name this variantPCS\-Densebecause it uses a denseSLC​\(T,L\)\\mathrm\{SLC\}\(T,L\)reward\. Figure[5](https://arxiv.org/html/2607.00485#Sx4.F5)shows the training curves\. Without the curriculum onτ\\tau, the policy’sSLC​\(T,L\)\\mathrm\{SLC\}\(T,L\)increases rapidly, but the accuracy underperforms PCS at every step and shows slight improvement over SoftLC RL\.

![Refer to caption](https://arxiv.org/html/2607.00485v1/x5.png)Figure 5:The training curves forSLC​\(T,L\)\\mathrm\{SLC\}\(T,L\)and Acc for PCS, PCS\-Dense, and SoftLC RL\.We attribute this to the inherently gradual nature of cross\-lingual transfer via code\-switched reasoning\. Because RL is optimized on mini\-batches, satisfying the language objective on the current batch does not necessarily generalize to other samples\. In contrast, dense rewards push the model to optimize language consistency too aggressively, encouraging premature language switching before the intermediate transfer stage is fully learned and thereby harming reasoning ability\.

### Sensitivity to the SLC Threshold

![Refer to caption](https://arxiv.org/html/2607.00485v1/x6.png)Figure 6:The SLC&Acc results of MMATH across differentτ\\tauused in evaluation\.We further examine whether our conclusions depend on the choice of the SLC threshold \(τ\\tauused in evaluation\) used in the SLC&Acc metric\. Specifically, we evaluate SLC&Acc under thresholds of 0\.80, 0\.85, 0\.90, and 0\.95 for SoftLC RL, M\-Thinker, and PCS\.

As shown in Figure[6](https://arxiv.org/html/2607.00485#Sx4.F6), PCS remains substantially more stable as the threshold is tightened\. Its SLC&Acc decreases only mildly, from 37\.9 at 0\.80 to 34\.3 at 0\.95\. SoftLC RL shows a similar but weaker trend, decreasing from 31\.6 to 27\.2\. In contrast, M\-Thinker drops much more sharply, from 38\.2 at 0\.80 to 22\.9 at 0\.95\. Notably, while M\-Thinker is competitive under relatively loose thresholds, PCS becomes clearly superior once stricter target\-language consistency is required\. These results show that the advantage of PCS is not tied to a particular threshold choice\. Instead, PCS is consistently more robust under increasingly strict language\-consistency requirements, suggesting that it induces more reliable target\-language reasoning rather than merely improving performance near a lenient cutoff\. This also supports our use of SLC≥\\geq0\.9 in the main results, which provides a practical balance between enforcing target\-language reasoning and tolerating language detection errors\.

## Related Work

### Multilingual Reasoning

Large reasoning models \(LRMs\) such as DeepSeek\-R1\(Guoet al\.[2025](https://arxiv.org/html/2607.00485#bib.bib3)\)and OpenAI o1\(OpenAIet al\.[2024](https://arxiv.org/html/2607.00485#bib.bib5)\)achieve strong complex reasoning via RLVR and test\-time scaling, yet in multilingual settings, they often exhibit an English bias\. DeepSeek\-R1 suffers from language mixing and mitigates this with cold\-start SFT and language\-consistency rewards\. However, directly forcing target\-language reasoning typically reduces accuracy\. Prior work tackles this issue through preference optimization \(MAPO\(Sheet al\.[2024](https://arxiv.org/html/2607.00485#bib.bib15)\)\), RL with language consistency and cross\-lingual thinking alignment \(MThinker\(Zhanget al\.[2026](https://arxiv.org/html/2607.00485#bib.bib11)\)\), and multilingual CoT\(Lai and Nissim[2024](https://arxiv.org/html/2607.00485#bib.bib17)\), training\-free structured reasoning\(Qiet al\.[2025b](https://arxiv.org/html/2607.00485#bib.bib18)\), and inference\-time representation steering\(Liet al\.[2025](https://arxiv.org/html/2607.00485#bib.bib19)\)\.

### Code\-Switching

Code\-switching, or language alternation, is a linguistic phenomenon where multilingual speakers use multiple languages within a conversation\(Poplack[1978](https://arxiv.org/html/2607.00485#bib.bib12)\)\. Code\-switching aids multilingual alignment, as demonstrated byLiet al\.\([2024](https://arxiv.org/html/2607.00485#bib.bib13)\), who use input\-only code\-switching during pre\-training\.Yooet al\.\([2024](https://arxiv.org/html/2607.00485#bib.bib14)\)introduces CSCL, a curriculum learning method using synthetic code\-switching data to enhance multilingual alignment\. SynCS\(Wanget al\.[2025b](https://arxiv.org/html/2607.00485#bib.bib1)\)analyzes how natural code\-switching enhances LLMs’ multilingual capabilities and proposes a more flexible and less expensive code\-switching synthesis approach\. Overall, code\-switching data can facilitate cross\-language transfer of models\.Sonet al\.\([2025](https://arxiv.org/html/2607.00485#bib.bib25)\)propose Language\-Mixed Chain\-of\-Thought, which permanently anchors reasoning in English while preserving target\-language terms at a fixed ratio, treating code\-switched reasoning as the desired end state rather than a transitional stage\.Onyameet al\.\([2026](https://arxiv.org/html/2607.00485#bib.bib26)\)employs code\-switching\-aware SFT followed by curriculum\-informed GRPO, but its curriculum schedules training over language resource tiers with a binary response\-level language reward, without explicitly driving the model toward fully target\-language reasoning\. In contrast, PCS treats code\-switching purely as an intermediate bridge and applies curriculum learning directly over the code\-switching ratio, progressively raising a step\-level language consistency threshold to eliminate code\-switching and ultimately achieve complete target\-language reasoning\. This design requires no distilled traces from stronger teachers or external judge models, and generalizes across typologically diverse languages\.

## Conclusion

We propose PCS \(Progressive Code\-Switching\), an effective method for transferring English reasoning ability to target languages without requiring distilled target\-language reasoning traces from stronger LRMs or supervision from a stronger judge model\. By initializing with code\-switched reasoning traces and progressively raising a step\-level language consistency threshold during RL, PCS smoothly shifts the model toward fully target\-language reasoning while maintaining task performance\. Experimental results across five languages demonstrate that PCS achieves the best Step\-Level Language Consistency and overall SLC&Acc\. Furthermore, our analysis confirms that it ensures a stable language transition and improves reasoning quality without reward hacking\.

## Limitations

Our study has several limitations\. First, due to limited computational resources, we primarily focus on standard long\-reasoning settings and do not explore substantially longer contexts \(e\.g\., 16K or 32K tokens\)\. It remains an open question whether PCS maintains the same language\-shifting stability and capability transfer behavior when the reasoning traces become much longer and require stronger long\-context generalization\. Second, our experiments are conducted with a relatively small backbone \(Qwen3\-4B\-Base 爱and Qwen3\-8B\-Base\)\. While PCS is model\-agnostic and does not depend on architecture\-specific assumptions, scaling to larger models may change the training dynamics and potentially improve the final multilingual reasoning performance\. We leave a systematic study of scaling effects—across both model size and context length—to future work\.

## Acknowledgments

We would like to thank the anonymous reviewers for their insightful comments\. Shujian Huang is the corresponding author\. This work is supported by National Science Foundation of China \(No\. 62376116\), the Fundamental Research Funds for the Central Universities \(No\. 2024300507\), Fundamental and Interdisciplinary Disciplines Breakthrough Plan of the Ministry of Education of China \(No\. JYB2025XDXM118\)\.

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## Appendix ADetailed Experiment Setup

### Data

For all training\-based methods in our experiments, we first perform a cold\-start SFT stage on the base models before RL training\. For the baseline methods, the cold\-start data consist of distilled target\-language reasoning traces, whereas for our method, they consist of code\-switched reasoning traces\.

For training\-based baselines, we largely follow the data recipe ofJiet al\.\([2025](https://arxiv.org/html/2607.00485#bib.bib10)\)\. The cold\-start SFT data for baselines are trained on 550K samples, including 500K English question–answer reasoning data fromTamet al\.\([2025](https://arxiv.org/html/2607.00485#bib.bib6)\)and 50K multilingual question–answer reasoning data distilled by DeepSeek\-V3\.2\-Exp \(10K per language\)\. The multilingual queries are randomly sampled from the 500K English data and translated to each language by DeepSeek\-V3\.2\-Exp\.

The cold\-start SFT data for PCS uses the same overall structure, except that the multilingual question–answer portion is replaced with code\-switched reasoning data: we randomly select 30% of the lines in the English reasoning traces and translate them into the target language using TranslateGemma\-4B\(Finkelsteinet al\.[2026](https://arxiv.org/html/2607.00485#bib.bib23)\)\.

For GRPO, we translate all math RL data fromTamet al\.\([2025](https://arxiv.org/html/2607.00485#bib.bib6)\)into five languages using DeepSeek\-V3\.2\-Exp and use them for all training\-based methods\.

### Training Settings

For SFT on both Qwen3\-4B\-Base and Qwen3\-8B\-Base, we train for 2 epochs with a global batch size of 128 and a learning rate of8×10−58\\times 10^\{\-5\}\. We pack training samples to a maximum sequence length of 32,768 tokens and set the sequence\-parallel size to 4\. For GRPO on both backbones, we train for 600 update steps with a learning rate of5×10−65\\times 10^\{\-6\}, a maximum response length of 4,096 tokens, and a training batch size of 256\. Training is fully on\-policy\. For each prompt, we sample 8 candidate responses during training, using a sampling temperature of 0\.9\. We use the same temperature \(0\.9\) at test time\. We follow the training settings of M\-thinker\(Zhanget al\.[2026](https://arxiv.org/html/2607.00485#bib.bib11)\)on our SFT model and using DeepSeek\-V3 as the judge model\. All experiments are conducted on 2x8 H100 GPUs\.

### Repetition Detection

We employ a multi\-level repetition detection mechanism to identify abnormal repetition patterns in model\-generated text\. Specifically, the detection approach comprises three main components:

- •Line\-level exact match detection: The text is split by line breaks and the frequency of each line’s content is counted\. A line is identified as repetitive when it repeats beyond a threshold \(≥\\geq20 times with length≥\\geq20 characters, or≥\\geq10 times with length≥\\geq50 characters\)\.
- •n\-gram sequence repetition detection: Based on suffix arrays and LCP \(Longest Common Prefix\) algorithms, this method uses divsufsort to construct suffix arrays, computes LCP arrays via the kasai algorithm, and employs sparse tables for efficient range queries to identify n\-grams of length≥\\geq2 with repetition counts≥\\geq20\.
- •Heuristic detection based on n\-gram statistics: This approach counts the frequency of fixed\-length n\-grams \(defaultn=20n=20\)\. When the highest\-frequency n\-gram appears≥\\geq20 times, it undergoes further validation using the suffix array method for precise verification\.

The detection process executes in order of priority, sequentially performing line\-level matching detection, n\-gram statistical detection, and suffix array verification\. This method effectively identifies various forms of text repetition, including both local vocabulary repetition and global structural repetition\.

## Appendix BMultilingual Alignment

![Refer to caption](https://arxiv.org/html/2607.00485v1/x7.png)Figure 7:The MEXA multilingual alignment score of PCS and baselines across model layers\.We examine whether PCS aligns its internal representations with English reasoning using the MEXA alignment score\(Kargaranet al\.[2025](https://arxiv.org/html/2607.00485#bib.bib21)\)\. Figure[7](https://arxiv.org/html/2607.00485#A2.F7)shows the MEXA alignment scores \(relative to English\) across layers\. Prior work\(Wendleret al\.[2024](https://arxiv.org/html/2607.00485#bib.bib22)\)suggests that LLMs typically rely on middle layers for reasoning, lower layers for task understanding, and upper layers for language realization\. PCS exhibits a level of alignment comparable to Naive RL—and higher than SLC RL—in the middle layers \(10–25\)\. Since Naive RL mainly performs reasoning in English, this result suggests that PCS preserves a stronger alignment with English\-style reasoning representations during the reasoning process\. In contrast, PCS achieves the lowest alignment scores in the upper layers, indicating larger divergence from English in language realization\. Overall, PCS maintains alignment with English reasoning while producing target\-language reasoning outputs\.

## Appendix CAnalysis of Potential Line\-Level Reward Hacking

Since PCS measures target\-language usage at theline\(step\) level, one potential failure mode is that the policy may pack many English reasoning tokens into a single long line, while keeping other lines in the target language, thereby satisfying the Step\-Level Language Consistency \(SLC\) constraint while retaining English\-style reasoning\.

To examine this possibility, we analyze PCS responses on MMATH\. For each response, we tokenize every line using Qwen3’s tokenizer\. We identify the longest line of each response \(measured in tokens\), and report the mean of this value over the evaluation set\. As shown in Table[3](https://arxiv.org/html/2607.00485#A3.T3), PCS indeed tends to produce longer longest\-lines than the baselines, suggesting that line length could be a confounding factor for line\-level language measurement\.

We further test whether the longest line contains substantial non\-target\-language content\. For each response, we apply \(i\)langdetectto the longest line to obtain a language label, and compute the language consistency \(LC\) as the percentage of longest lines classified as the target language; and \(ii\) we use DeepSeek\-V3 as a detector to estimate the code\-switching ratio within the longest line\. The prompt is detailed at Table[C](https://arxiv.org/html/2607.00485#A3), and we report the mean of the code\-switching label \(0/1\) over the evaluation set\. Table[4](https://arxiv.org/html/2607.00485#A3.T4)shows that the longest lines are overwhelmingly classified as the target language bylangdetect\(LC≥\\geq96\.7% across all languages\)\. Moreover, the estimated intra\-line code\-switching ratio is very low \(0\.7–1\.6%\), indicating that PCS does not appear to hide large amounts of English reasoning inside the longest line\. Overall, while PCS shows a tendency to generate longer lines, we find no evidence that it exploits this behavior to bypass the line\-level SLC constraint\. These results support the reliability of our line\-level language measurement in practice\.

Code\-Switching Detecting Prompt\{text\}Please determine whether the language of the given text is consistent, meaning it contains only a single language \(excluding mathematical content\)\.Return 0 if it contains only one language, and 1 if it mixes multiple languages\. Output exactly 0 or 1 with no additional text or explanations\.

![Refer to caption](https://arxiv.org/html/2607.00485v1/x8.png)Figure 8:Instructions used for Prompt ControlMethodFRPTJAKOTHNaive RL178169181173201SoftLC RL184160169139197M\-Thinker145140145158183PCS223220193183195Table 3:Average token length of the longest line of PCS’s response on MMATH\.StatisticsFRPTJAKOTHLC\(%\)99\.198\.898\.796\.799\.3CS Ratio\(%\)0\.71\.61\.21\.21\.5Table 4:Language Consistency \(LC, detect using langdetect tool\) and Code\-Switching Ratio \(CS Ratio, detect using DeepSeek\-V3\) of the longest line of PCS’s response on MMATH\.
## Appendix DReference Update and KL

![Refer to caption](https://arxiv.org/html/2607.00485v1/x9.png)Figure 9:The ablation results for PCS and PCS without reference\-model reset and KL regularization\.As described in Section[Progressive Code\-Switching RL](https://arxiv.org/html/2607.00485#Sx2.SSx4), we reset the reference model and apply KL regularization when the targetSLC​\(T,L\)\\mathrm\{SLC\}\(T,L\)become large\. The reason is that, in the late stage of training, the model is encouraged to produce almost entirely target\-language reasoning, making the policy more prone to drifting away from the earlier policy\. If left unconstrained, this drift can destabilize RL optimization and hurt final performance\. Reference\-model reset mitigates this mismatch by updating the optimization anchor to a policy that is better aligned with the current transfer stage, while KL regularization keeps subsequent updates controlled\. As shown in Figure[9](https://arxiv.org/html/2607.00485#A4.F9), removing these components leads to a substantial performance drop, highlighting their role in stabilizing progressive language transfer\.

## Appendix EReward Hacking for Response\-Level Language Consistency

As demonstrated in Figures[10](https://arxiv.org/html/2607.00485#A5.F10)and[11](https://arxiv.org/html/2607.00485#A5.F11), when prompted with a Thai math question, M\-Thinker’s response exhibits extensive token\-level code\-switching between English and Chinese\. However, when we apply thelangdetecttool to identify the response\-level language, it returns “\[th:0\.9999999495640537\]”, indicating that the response is entirely Thai, which is clearly not the case\. Although M\-Thinker produces the correct answer, it relies on English and Chinese reasoning abilities andhacksthe response\-level language detector in a highly unreadable manner\.

![Refer to caption](https://arxiv.org/html/2607.00485v1/x10.png)Figure 10:Case Study of M\-Thinker’s model for one sampled Thai math question\. The language of this response detected by langdetect is: \[th:0\.9999999495640537\]![Refer to caption](https://arxiv.org/html/2607.00485v1/x11.png)Figure 11:Case Study of PCS’s model for one sampled Thai math question\.