Paper deep dive
Which Language Should a Skeleton Speak? Language Choices in Multilingual Reasoning
HyeonSeok Lim, SeungWoo Song, Inho Won, Hoyun Song, Jihyo Kim, KyungTae Lim
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 93%
Last extracted: 10/11/2026, 2:52:47 AM
Summary
This paper introduces the Language-Aware Skeleton Exploration Framework (LASEF) to investigate the impact of skeleton language choices in multilingual mathematical reasoning. The study finds that while English skeletons generally offer small average performance gains, particularly for smaller models and low-resource languages, these gains are often not statistically significant after correction. The authors identify three patterns of language effects: directionally consistent, benchmark-dependent, and asymmetric negative, concluding that skeleton language is a context-dependent variable rather than a universally optimal choice.
Entities (9)
Relation Signals (6)
Qwen2.5-7B → evaluatedon → MGSM
confidence 95% · For instance, with QWEN2.5-7B on MGSM (CoT-en † ), Swahili performance rose from 63.1 to 71.7
Swahili → isclassifiedas → Low-Resource Language
confidence 95% · Gains are substantially larger for low-resource languages like Swahili (sw)
Skeleton-based reasoning → istrainingfree → true
confidence 95% · A key property of skeleton-based reasoning is its training-free nature
LASEF → proposesstudyof → skeleton-language choice
confidence 95% · We propose the Language-Aware Skeleton Exploration Framework (LASEF) to study skeleton-language choice in multilingual mathematical reasoning.
English skeletons → yieldsperformancegainfor → Smaller Models
confidence 90% · English skeletons yield a small positive tendency on average, most visible for smaller models and low-resource languages.
English skeletons → yieldsperformancegainfor → Low-resource languages
confidence 90% · English skeletons yield a small positive tendency on average, most visible for smaller models and low-resource languages.
Cypher Suggestions (0)
No Cypher suggestions yet.
Abstract
Abstract:Skeleton-based reasoning prompting is a promising training-free approach for structuring LLM reasoning, but prior work largely assumes an English-centric setting. We propose the Language-Aware Skeleton Exploration Framework (LASEF) to study skeleton-language choice in multilingual mathematical reasoning. Across math benchmarks, model scales, and languages, we show that English skeletons yield a small positive tendency on average, most visible for smaller models and low-resource languages. However, few language-level gains remain significant after correction, and English is not universally optimal. Combining greedy decoding, multi-rollout evaluation, translation ablation, and cross-benchmark validation, we further find three patterns of skeleton-language effects: directionally consistent, evaluation- and benchmark-dependent, and asymmetric negative. These effects cannot be fully explained by generation quality alone. Overall, skeleton language is a context-dependent design variable that requires multi-level exploration. All resources are released at this https URL.
Tags
Links
- Source: https://arxiv.org/abs/2610.09607v1
- Canonical: https://arxiv.org/abs/2610.09607v1
Trouble viewing inline? Open PDF directly →
Full Text
103,175 characters extracted from source content.
Expand or collapse full text
Which Language Should a Skeleton Speak? Language Choices in Multilingual Reasoning HyeonSeok Lim 1 , SeungWoo Song 2 , Inho Won 2 , Hoyun Song 2,3 , Jihyo Kim 2 KyungTae Lim 2† 1 ETRI 2 KAIST 3 Department of Artificial Intelligence, Dankook University gustjrantk@etri.re.kr, sswoo,kotmul2no,jihyo.kim,ktlim@kaist.ac.kr, hoyun.song@dankook.ac.kr Abstract Skeleton-based reasoning prompting is a promising training-free approach for structur- ing LLM reasoning, but prior work largely assumes an English-centric setting. We pro- pose the Language-Aware Skeleton Explo- ration Framework (LASEF) to study skeleton- language choice in multilingual mathematical reasoning. Across math benchmarks, model scales, and languages, we show that English skeletons yield a small positive tendency on av- erage, most visible for smaller models and low- resource languages. However, few language- level gains remain significant after correction, and English is not universally optimal. Com- bining greedy decoding, multi-rollout evalua- tion, translation ablation, and cross-benchmark validation, we further find three patterns of skeleton-language effects: directionally con- sistent, evaluation- and benchmark-dependent, and asymmetric negative. These effects can- not be fully explained by generation quality alone. Overall, skeleton language is a context- dependent design variable that requires multi- level exploration. All resources are released 1 . 1 Introduction Recent studies on structured reasoning prompting have employed an approach (Wang et al., 2023; Khot et al., 2022). This approach guides a model’s reasoning process by concisely presenting the core flow of thought required for problem solving. In this paper, we refer to such explicit representations of reasoning flow as a skeleton. A skeleton provides a high-level indication of the direction or strategy for solving a given input query, upon which more detailed reasoning and the final answer are subse- quently generated (Ning et al., 2023; Li et al., 2024; Qi et al., 2025c). A key property of skeleton-based reasoning is its training-free nature, which enables its use in low-resource language scenarios. † Corresponding author. 1 https://github.com/lhsstn/LASEF However, when applying skeletons in multilin- gual settings, there exists a fundamental design challenge. A skeleton-based reasoning pipeline im- plicitly involves three language choices: (1) the language of the input query, (2) the language in which the skeleton is expressed, and (3) the lan- guage in which reasoning and the final answer are generated. Despite the fact that different combina- tions of these elements may lead the model to rely on different linguistic priors and reasoning trajec- tories (Wang et al., 2025a, 2024), prior work has not systematically explored this design space of language combinations. Prior work on skeleton-based reasoning has largely assumed an English-centric setting (Zhou et al., 2022; Qi et al., 2025c), focusing on how to design the structure or prompt format of skeletons to improve performance. While some studies have applied skeletons to multilingual or cross-domain scenarios (Qin et al., 2023; Huang et al., 2023), most of them either fix the skeleton language to English or treat language choice as a secondary factor. As a result, the crucial question of which language should be utilized for the reasoning skele- ton remains insufficiently analyzed for real-world multilingual deployment. Meanwhile, recent prior studies have reported that even the same model can exhibit different per- formance depending on the language in which rea- soning is carried out (Qi et al., 2025b), and have shown that performing reasoning after translating the query into English can be effective (Shi et al., 2022; Liu et al., 2025; Zhu et al., 2024). These find- ings suggest that skeletons are not merely formal tools, but may function as a medium that activates language-specific reasoning priors formed during pretraining (Etxaniz et al., 2024). At the same time, they also imply that the effectiveness of skeletons may not be uniform across languages (Qian et al., 2024; Chen et al., 2024). In this context, we pose the following research arXiv:2610.09607v1 [cs.CL] 7 Oct 2026 question: Under which linguistic conditions does skeleton-based reasoning operate effectively, and how do different combinations of input language, skeleton language, and reasoning/answer language affect reasoning performance? In particular, how do these effects differ in low-resource languages? This question goes beyond asking “which skeleton is effective,” and instead addresses a more funda- mental design issue: which language should be used for the skeleton. To investigate this question, we propose Language-Aware Skeleton Exploration. Here, ex- ploration does not involve any new algorithmic learning or parameter updates; instead, it refers to an empirical approach that systematically analyzes performance variations across different language combinations while keeping the skeleton structure and format fixed, with the goal of identifying effec- tive language configurations. This setup allows us to disentangle the impact of skeleton design itself from that of language choice. Our study focuses on mathematical reasoning, a domain where multilingual performance gaps are known to be significant due to limited reasoning capabilities in non-English languages (Qi et al., 2025b). We conduct experiments across bench- marks of varying difficulty levels, models of differ- ent scales, and a wide range of language settings spanning high-resource to low-resource languages. Through this comprehensive evaluation, we system- atically analyze how the effectiveness of skeletons varies across languages and conditions. Building on these findings, we derive practical insights for the reliable use of skeleton-based reasoning in multi- lingual environments. Our results answer the above question conditionally rather than universally: in- stead of prescribing a single optimal skeleton lan- guage, we characterize when skeleton-language choices help or hurt and how consistently these effects generalize across decoding protocols and benchmarks. In summary, the contributions of this paper are as follows: •We formalize the combination of input, skele- ton, and reasoning languages as a core design variable in multilingual skeleton-based reason- ing, and propose a training-free framework (LASEF) to analyze this design space. • Combining greedy decoding and multi-rollout evaluation across math benchmarks of varying difficulty, we demonstrate the non-uniform ef- fectiveness of skeleton languages and assess the directional consistency of the observed effects across decoding protocols and bench- marks. •We identify three distinct patterns of skeleton- language effects: directionally consistent, evaluation- and benchmark-dependent, and asymmetric negative. These patterns are sup- ported by a translation ablation that disentan- gles skeleton generation quality from cross- lingual alignment. 2 Related Work English-PivotReasoning WhileChain-of- Thought (CoT) prompting (Wei et al., 2022) has enhanced LLM reasoning, a significant performance gap persists in non-English contexts, particularly in mathematical tasks (Shi et al., 2022; Ahuja et al., 2023; Wang et al., 2025c). To mitigate this disparity, English-pivot strategies that translate inputs or reasoning processes into English have been proposed to leverage the models’ strong English priors (Qin et al., 2023; Huang et al., 2023; Liu et al., 2025; Etxaniz et al., 2024). However, this approach often sacrifices interpretability for non-English users by generating final answers in English, thereby limiting practical utility in native-language applications (Wang et al., 2025b; Lee et al., 2025; Qi et al., 2025b). Structured Reasoning Research on structure- aware prompting has evolved from Zero-shot CoT (Kojima et al., 2022) to more sophisticated ap- proaches such as Skeleton-of-Thought (Ning et al., 2023), Plan-and-Solve (Wang et al., 2023), and Program-of-Thought (Chen et al., 2022). These methods have demonstrated that explicitly structur- ing reasoning enhances logical consistency (Zhou et al., 2022; Yao et al., 2023; Khot et al., 2022). While efforts have been made to extend structured reasoning to multilingual contexts through cross- lingual instruction tuning (Chai et al., 2025; Lai and Nissim, 2024), prior work has predominantly fixed the skeleton language to English, overlooking the impact of language combinations between the structural guide and the final response (Qi et al., 2025c; Li et al., 2024). In this study, we refer to such structured guides as a skeleton and adopt the inference structure of Qi et al. (2025c) as a baseline to ensure experimental control. By fixing the structure and manipulating only the skeleton’s language, we systematically analyze how language configuration affects multi- lingual reasoning performance. 3 Language-Aware Skeleton Exploration Framework This study investigates how the effectiveness of skeleton-based reasoning varies depending on lan- guage choice and evaluation contexts within a mul- tilingual environment. To this end, we propose the Language-Aware Skeleton Exploration Framework (LASEF), as illustrated in Fig. 1, which systemati- cally analyzes skeleton utilization by varying lan- guage configurations while keeping the structure and format of the skeleton fixed. Here, “exploration” does not refer to new training or algorithmic search, but rather to an empirical analysis process designed to identify effective language configurations within a given evaluation environment. 3.1Problem Setup and Language Dimensions Standard language model inference typically in- volves a queryqand a corresponding answera, where the query is presented in a specific language ℓ q . In skeleton-based reasoning, the model first gen- erates a skeletons, summarizing the core structure required for problem-solving based on the query, and subsequently uses it to generate the detailed reasoning process and the final answer. This pro- cess entails three distinct language choices given a user’s target languageℓ t : (1) the language of the query,ℓ q ; (2) the language of the skeleton,ℓ s ; and (3) the language in which the detailed reasoning and final answer are generated, ℓ a . These three languages may be identical or dis- tinct; depending on their combination, the linguistic priors and reasoning paths elicited by the model can vary. Thus, a single skeleton-based reasoning process is characterized by the language combina- tion(ℓ q ,ℓ s ,ℓ a ). LASEF focuses on isolating and analyzing the impact of these language configura- tions on reasoning performance while fixing the skeleton generation method and structure. Under this formulation, the skeleton-based reasoning pro- cess can be expressed as: ˆa = f q (ℓ q ) ,s (ℓ s ) ,ℓ a , wheref (·)represents the inference function of the pre-trained language model, ands (ℓ s ) denotes the skeleton expressed in language ℓ s . Notably, the effective language combination for skeleton utilization is not universally fixed but may vary depending on the benchmark or domain used for evaluation. Given a specific benchmarkBcon- sisting of query-answer pairs(q,a∈ B), LASEF provides an analytical framework to identify rel- atively effective skeleton language configurations within that evaluation environment by comparing performance variations under different language combinations (ℓ q ,ℓ s ,ℓ a ). This problem setup assumes a training-free en- vironment involving no additional training or pa- rameter updates. The following section describes the specific language configurations and skeleton usage methods considered in this study. 3.2 Skeleton Construction and Usage The skeletons used in this study are intermediate representations that concisely reveal the reasoning structure for problem-solving, rather than present- ing core answers or calculation procedures. In other words, the skeleton provides a minimal structural guide for the reasoning process to follow, avoiding specific instructions for the actual solution steps. This perspective conceptually aligns with existing research interpreting skeletons as structural repre- sentations that organize reasoning, rather than as summaries or hints for the solution. Accordingly, our skeletons maintain a very short and abstract form, excluding specific calculation steps or answers. To ensure this, we consistently apply the following principle during skeleton gener- ation: “Provide only the essential structural charac- teristics of the problem. Describe what the problem structure is, never how to solve it.” Fig. 1 presents a representative example of a skeleton generated in English (ℓ s = en) for a Swahili query (ℓ q = sw). Adhering to our pro- posed criteria, skeletons maintain a concise length of approximately 120 tokens on average regardless of the language, devoid of direct answers or core solution steps. We provide a detailed quantitative analysis of skeleton length and content distribu- tion, along with the complete generation rules and additional examples, in Appendix A. 4 Experiments In this section, we systematically validate the effec- tiveness of the proposed Language-Aware Skeleton Exploration Framework (LASEF) across diverse settings. The experiments are designed to analyze performance by varying model scale, problem diffi- culty, and language resource levels, while keeping (A) Input Query User provides 푞 (ℓ ! ) ℓ $ : query language (B) Skeleton Generation Generate skeleton 푠 (ℓ " ) from 푞 (ℓ ! ) Fixed skeleton format; vary only ℓ % ℓ % : skeleton language (C) Skeleton-Guided Reasoning Generate reasoning and answer in ℓ & 푎̂←푓(푞 ℓ ! ,푠 (ℓ " ) ,ℓ & ) ℓ & : answer language (reasoning & answer) (D) Language Configurations ℓ ' : TargetLanguage(e.g., Swahili, ..) Design variables: (ℓ $ ,ℓ % ,ℓ & ) Examples (used in experiments) (E) Benchmark-Conditioned Analysis Evaluate configurations on benchmark B Training-free: no parameter updates Identify effective configurations for B query query skeleton configs outputs [Swahili]Jimmy ana $2 zaidiyamara mbiliyapesaza Ethel. IwapoEthel ana $8, Jimmy ana pesangapi? (Jimmy has twice as much money as Ethel plus $2. If Ethel has $8, how much money does Jimmy have?) :Examples :LLM-based generation modules CoT- ℓ ! : CoT- ℓ ! +Skeleton : CoT- 푒푛: CoT- 푒푛+Skeleton : ℓ ! ,−,ℓ ! ℓ ! ,푒푛,ℓ ! 푒푛,−,ℓ ! 푒푛,푒푛,ℓ ! Key Concepts / Tools -Algebraic expressions -Linear relationships Problem Structure -Known: Ethel's money amount ... -Unknown: Jimmy's money ... Figure 1: Overview of the Skeleton-Guided Reasoning Framework. (A) The process begins with an input queryq (ℓ q ) . (B) A structured skeletons (ℓ s ) is generated to abstract the problem logic. (C) The final reasoning and answerˆaare produced in the answer languageℓ a , conditioned on the query and skeleton. (D) We define the language configuration space as(ℓ q , ℓ s , ℓ a )and compare four primary strategies: standard CoT and skeleton-guided approaches, applied in both the target language (ℓ t ) and English (en). (E) Effective configurations are identified through training-free benchmark analysis. the structure and generation method of the skeleton fixed. Through this, we analyze (1) whether the effectiveness of skeleton-based reasoning is lim- ited to specific conditions, and (2) how it varies according to language and model characteristics. 4.1 Experimental Setup Models based on Scale.To analyze the impact of model scale, we select the QWEN2.5 series (Qwen et al., 2025) as our base models, following prior research (Qi et al., 2025c). These models are cho- sen for their superior multilingual capabilities as general-purpose models, rather than being special- ized solely for reasoning. We evaluate 7B, 14B, and 72B models to examine how skeleton guid- ance scales. For brevity, we refer to the QWEN2.5- INSTRUCT models by their sizes (e.g., QWEN2.5- 14B). Results for the LLAMA 3.1 series (Grattafiori et al., 2024) are provided in Appendix G. Benchmarks by Difficulty. We categorize our benchmarks by difficulty to assess skeleton ef- fectiveness across task complexities. We employ MGSM (Shi et al., 2022) for elementary arith- metic, MATH-500 (Lightman et al., 2023) for intermediate-level problems, and PolyMath (Wang et al., 2025c) for advanced reasoning tasks requir- ing complex structural analysis. Target Languages based on Resource Levels. To evaluate the impact of skeletons according to language resource levels, we included diverse tar- get languages, from high-resource to low-resource. High-resource languages were selected based on their sufficient exposure during large-scale pre- training, while low-resource languages were cho- sen based on the relative scarcity of training data, which often leads to inconsistent reasoning per- formance in multilingual LLMs. This allows us to analyze whether skeletons play different roles depending on the resource level of the language. Evaluation Protocol.We use Exact Match (EM) as the evaluation metric, restricted to samples sat- isfyingℓ a = ℓ t to ensure multilingual reasoning validity; reported scores are therefore conditional reasoning accuracies on language-compliant sam- ples. Alongside greedy decoding, we conduct multi- rollout evaluation (T =0.7,N =5) to assess the sta- bility of skeleton-language effects. Details on lan- guage identification, filtering and retention rates, and pairing are in Appendix B, and unfiltered re- sults are in Appendix L. 4.2 Baseline Configurations To analyze the effectiveness of LASEF, we de- fine the language combination(ℓ q ,ℓ s ,ℓ a )for a given target languageℓ t , and establish an appro- priate comparative baseline. Since skeletons func- tion as structural guides for improving reasoning performance, we adopt a Chain-of-Thought (CoT)- based setting as the primary baseline (Wei et al., 2022), following prior work on multilingual rea- soning (Huang et al., 2023; Qin et al., 2023). Target-Language CoT (COT-ℓ t ).In this setting, the input query is given in the target language, and both reasoning and the answer are generated in the Method LanguageMGSMMATH-500PolyMath (ℓ q , ℓ s , ℓ a )zheskothswAVG.zheskothswAVG.zheskothswAVG. Qwen2.5-7B-Instruct COT-ℓ t ℓ t ,−, ℓ t 79.980.769.780.216.065.356.457.547.849.814.245.130.831.026.426.15.123.9 + SKELETONℓ t , en, ℓ t 80.783.871.079.022.967.555.457.749.351.920.446.930.632.128.027.67.425.1 COT-en † en,−, ℓ t 82.384.374.578.163.176.556.057.451.649.524.447.829.732.127.529.820.327.9 + SKELETONen, en, ℓ t 82.385.973.278.971.778.457.458.252.353.829.850.332.034.427.730.419.928.9 Qwen2.5-14B-Instruct COT-ℓ t ℓ t ,−, ℓ t 88.890.077.987.645.077.958.362.654.757.430.152.631.836.731.630.914.729.1 + SKELETONℓ t , en, ℓ t 88.090.080.785.950.779.159.464.056.959.935.255.134.136.932.233.315.030.3 COT-en † en,−, ℓ t 83.188.381.284.874.5 82.458.860.253.855.139.153.432.336.530.132.024.031.0 + SKELETONen, en, ℓ t 86.887.982.584.874.583.359.662.255.255.840.454.633.735.432.232.026.932.0 Qwen2.5-72B-Instruct COT-ℓ t ℓ t ,−, ℓ t 88.391.683.990.868.084.563.163.863.261.646.359.636.137.534.534.731.034.8 + SKELETONℓ t , en, ℓ t 88.089.284.789.670.184.365.863.862.463.454.361.937.137.537.035.032.035.7 COT-en † en,−, ℓ t 85.591.582.786.385.586.362.264.463.161.756.361.534.235.835.335.840.636.3 + SKELETONen, en, ℓ t 86.890.783.186.786.786.864.264.061.162.560.062.434.037.936.233.139.336.1 Table 1: Results on MGSM, MATH-500, and PolyMath.(ℓ q , ℓ s , ℓ a )denote the query, skeleton, and answer languages, respectively. CoT-en † indicates queries translated into English using Google Translate, and the skeleton language is fixed to English unless otherwise noted. ‘AVG.’ denotes the average over five languages (zh, es, ko, th, sw). Greedy decoding throughout. same language(ℓ q = ℓ a = ℓ t ). We use a standard CoT prompt (“Let’s think step by step, Respond in ℓ t ”) without applying a skeleton. This represents the most natural usage scenario where users verify both the query and the reasoning process in their native language. English-Pivot CoT (COT-en). In this setting, the input query is translated into English(ℓ q = en), after which the reasoning and answer are generated in the target languageℓ t . This baseline is essen- tial for two reasons. First, comparing performance based on English-translated queries is a widely used practice in existing multilingual reasoning studies (Qin et al., 2023; Huang et al., 2023; Liu et al., 2025), representing the upper-bound perfor- mance of LLMs pre-trained primarily on English. Second, even in real-world non-English user envi- ronments, since users already understand the mean- ing of the query, internally converting the query to English does not hinder meaning transmission. Thus, COT-enserves as a rational baseline to eval- uate whether performance gains via translation are achievable without structural guidance. Skeleton Baseline Language Configuration. English skeletons are used in Tab. 1 and 2, while Fig. 2 and 3 expands to Chinese, Spanish, Russian, Korean, and Thai. 4.3 Performance across Diverse Dimensions Tab. 1 compares LASEF in an English skeleton setting(ℓ q ,en,ℓ a )against baselines across three benchmarks: MGSM, MATH-500, and PolyMath. Bridging the Resource GapLanguage resource levels significantly influence the impact of English skeletons. Gains are substantially larger for low- resource languages like Swahili (sw) compared to high-resource languages like Chinese (zh) and Spanish (es). For instance, with QWEN2.5-7B on MGSM (CoT-en † ), Swahili performance rose from 63.1 to 71.7 (+8.6), while Chinese and Span- ish showed marginal gains (+0.0 and +1.6). This trend persists across scales; with QWEN2.5-72B on MATH-500, Swahili again saw the highest im- provement (+3.7). These results indicate that when low-resource languages lack sufficient reasoning stability, English skeletons serve as an essential scaffold to stabilize the process. Does Scaling Always Guarantee Improvement? With respect to model scale, we observe that smaller models benefit more from English skele- tons, while the effect diminishes or even becomes negative as model size increases. QWEN2.5-7B and 14B exhibit performance improvements after applying skeletons across all benchmarks (+0.9 to 2.5). In contrast, QWEN2.5-72B shows an aver- age performance drop of 0.2 after skeleton applica- tion under the COT-ℓ t setting on MGSM and the COT-en † setting on PolyMath. This suggests that for large-scale models, externally imposed English skeletons may act as an unnecessary constraint rather than a helpful inductive bias. Does Benchmark Difficulty Limit the Gains? Examining performance variations across bench- mark difficulty levels, we observe that the contri- bution of skeletons is larger for easier benchmarks. Using QWEN2.5-7B as the reference model, skele- tons yield a performance gain of +2.2 on MGSM, the easiest benchmark, compared to +1.8 on MATH-500, which has medium difficulty, and +1.2 Language ℓ t ,en,ℓ t en † ,en,ℓ t COT-l t +SKELETONCOT-en † +SKELETON Qwen2.5-7B-Instruct k40.0851.42 (+11.34)62.8272.22 (+9.40) ky33.3337.86 (+4.53)61.9265.69 (+3.77) mn27.1633.33 (+6.17)66.9664.78 (-2.18) ug27.1337.65 (+10.52)57.1460.37 (+3.23) hy50.8156.05 (+5.24)64.4973.88 (+9.39) lo25.5123.48 (-2.03)48.5448.95 (+0.41) eu20.7619.92 (-0.84)65.6274.55 (+8.93) mt25.6520.00 (-5.65)62.1158.59 (-3.52) am8.2310.29 (+2.06)39.6332.93 (-6.70) ne54.2952.65 (-1.64)74.5878.81 (+4.23) si10.1612.20 (+2.04)37.1435.92 (-1.22) ps26.4626.91 (+0.45)37.6247.03 (+9.41) sd20.2722.07 (+1.80)64.8168.52 (+3.71) pa51.4152.21 (+0.80)55.8752.63 (-3.24) gu44.3547.98 (+3.63)59.8460.64 (+0.80) mr45.3446.96 (+1.62)75.4174.18 (-1.23) or44.3543.95 (-0.40)36.8438.06 (+1.22) ta47.7949.80 (+2.01)69.4869.08 (-0.40) ml42.7446.37 (+3.63)62.2564.66 (+2.41) AVG.33.9936.37 (+2.38)58.0660.08 (+2.02) Table 2: Performance comparison on 19 low-resource lan- guages (MGSM) using QWEN2.5-7B with English Skele- ton. CoT-en † : English-translated queries via Google Translate. Parentheses show ∆ gains. on PolyMath, the most challenging benchmark. This trend suggests that as problem difficulty in- creases, simple skeleton generation alone becomes insufficient, as harder problems require more so- phisticated and precise computational reasoning beyond what skeletons can provide. Summary of Results.Results show that English skeletons’ benefits depend on model scale, task difficulty, and language resources. Gains are most pronounced for smaller models, easier tasks, and low-resource languages. Notably, the CoT-en † set- ting consistently outperforms CoT-ℓ t , reaffirming the strong English prior of LLMs. This underscores translation as a critical baseline. In low-resource settings, most of the improvement comes from translation rather than from skeletons (+24.07 vs. +2.02 p in Tab. 2). 4.4 Effect of English Skeletons in Low-Resource Languages Previous analysis confirms that English skeletons are most effective for smaller models, easier bench- marks, and low-resource languages. In this section, we validate this finding by evaluating QWEN2.5- 7B on 19 low-resource MGSM languages. This section examines the consistency of skeleton-based CoT improvements under these conditions. The criteria for language selection and our translation methodology are detailed in Appendix C. Tab. 2 summarizes the performance changes under the target-language input (ℓ t ) and English- translated input (en † ) settings. English skeletons show a small positive tendency on average: apply- ing them improves 13 of the 19 languages, with mean gains of +2.38 in the CoT-ℓ t setting and +2.02 in the CoT-en † setting. However, after FDR correction, statistically significant language-level improvements are observed only for Kazakh (k) and Uyghur (ug); pooling the 19 languages at the item level, the overall effect remains significant (Appendix J). First, under the Native Input (CoT-ℓ t ) setting, skeletons substantially unlock the model’s latent reasoning capacity for certain languages, such as Kazakh (k) (+11.34) and Uyghur (ug) (+10.52). However, there also exist cases where performance degrades, such as Maltese (mt) (5.65), indicating a high variance in language-specific effects. Second, under the (CoT-en † ) setting, the ef- fect of English skeletons becomes considerably more consistent in direction. Notably, query trans- lation alone leads to a dramatic increase in av- erage performance, from 33.99 to 58.06. When English skeletons are incorporated, an additional performance gain of +2.02 is observed. In partic- ular, languages such as Armenian (hy) (+9.39), Basque (eu) (+8.93), and Pashto (ps) (+9.41) ex- hibit substantial improvements when structure in the form of skeletons is added on top of translated queries. Overall, in low-resource language settings, query translation provides the dominant improvement, and English skeletons contribute a smaller addi- tional gain by structuring the reasoning process. This suggests that skeleton-based CoT can serve as a useful supplementary reasoning aid—rather than the main driver of improvement—even in ex- tremely low-resource language scenarios. 4.5 Is English the Only Solution for Low-Resource Languages? In Sec. 4.4, we showed that English skeletons gen- erally serve as effective reasoning aids. However, the COT-ℓ t results in Tab. 2 indicate that English skeletons are not always beneficial for all low- resource languages. In particular, five out of the 19 languages (lo, eu, mt, ne, and or) show perfor- mance degradation when English skeletons are ap- plied (the 19 languages are those of the 34 MGSM low-resource languages that pass the sample filter; see Appendix B.2). ZHESRUKOTH Skeleton Language K KY MN UG HY LO EU MT AM NE SI PS SD PA GU MR OR TA ML AVG CoT Language (Grouped) -3.31-2.020.00-7.66-4.05 -2.502.080.00-6.97-0.81 -4.56-5.53-5.00-0.88-8.47 -6.45-4.53-4.11-6.06-10.25 -4.421.63-2.035.17-6.50 3.63-2.801.602.472.81 1.72-2.030.006.320.48 3.756.618.621.886.90 -2.08-0.44-2.071.71-1.70 2.50-1.23-0.83-0.90-0.87 -0.40-0.81-0.41-2.21-1.23 -1.73-2.650.440.00-0.87 -3.29-4.21-1.04-1.170.54 -1.61-3.61-3.61-1.04-4.82 -0.80-4.40-5.26-5.04-4.00 3.641.622.050.68-0.41 0.40-1.213.72-1.22-3.24 0.00-5.20-2.40-6.60-4.42 2.81-2.41-1.63-1.28-4.02 -0.67-1.64-0.63-1.20-2.36 Greedy Decoding Average Skeleton Delta (Non-English English) (a) Greedy Decoding (single rollout) ZHESRUKOTH Skeleton Language K KY MN UG HY LO EU MT AM NE SI PS SD PA GU MR OR TA ML AVG CoT Language (Grouped) -4.54-2.05-1.57-6.84-5.26 -3.97-0.45-5.06-6.06-4.23 -1.81-1.62-2.65-2.39-4.70 -1.74-4.29-3.82-5.95-7.98 -2.50-2.11-1.344.61-1.57 0.74-1.48-0.460.233.18 1.232.451.282.78-0.93 -0.44-0.98-0.24-2.81-1.79 -2.60-1.64-0.710.070.12 1.361.160.85-0.95-0.72 0.050.340.241.680.50 -2.80-1.100.15-2.30-2.74 -3.70-2.51-3.04-1.08-4.13 -1.28-2.83-3.13-5.47-4.06 -2.480.32-3.24-1.09-2.38 0.46-1.16-2.17-1.03-0.99 -0.900.381.07-0.40-4.25 -0.76-5.84-3.18-5.28-4.42 0.08-0.72-0.57-2.17-3.04 -1.35-1.27-1.45-1.81-2.60 5-Rollout Average Skeleton Delta (Non-English English) (b) Multi-rollout (T =0.7, N =5) Figure 2: Performance differences (Non-English−English) on MGSM under CoT-ℓ t with QWEN2.5-7B. The English skeleton is replaced with five Non-English skeletons (ZH, ES, RU, KO, TH); the same skeleton set is used in both panels. (a) Greedy decoding (single rollout). (b) Multi-rollout sampling (T =0.7,N =5). The corresponding English-skeleton reference values (English skeleton vs. no skeleton per language) are reported in Tab. 2. In this section, we extend LASEF to five non- English skeletons (zh, es, ru, ko, and th) and an- alyze how the effect of skeleton language choice varies across target languages. We distinguish sur- face affinity (language family, script, and geo- graphic proximity) from structural alignment (ty- pology and word order), and examine the stability of the observed effects using both greedy decoding and multi-rollout sampling. General Trends: English as a Stable Anchor. Fig. 2 shows the performance differences between non-English and English skeletons under the COT- ℓ t setting with QWEN2.5-7B, using greedy decod- ing and multi-rollout evaluation. Overall, apply- ing non-English skeletons uniformly leads to an average performance decrease across languages, ranging from 0.63 to 2.60 points. This result reaf- firms that English serves as a stable reasoning skele- ton for most languages. However, when examining individual languages, notable exceptions emerge, particularly for languages on which the English skeleton had a negative effect. Directionally Consistent Cross-Lingual Effects. As shown in Tab. 2, some languages that experience performance degradation with English skeletons ex- hibit clear improvements when paired with alterna- tive skeletons that share affinity or alignment with the target language. Lao (lo) suffers a performance drop with the English skeleton (−2.03), but shows a+2.81improvement with a Thai skeleton, which shares surface affinity through the Tai–Kadai lan- guage family (Fig. 2(a)). Similarly, Basque (eu) and Armenian (hy) achieve gains of +6.32 and +5.17, respectively, when paired with a Korean skeleton, which shares their structural alignment (aggluti- native morphology and SOV word order). Cru- cially, the sign of these effects is preserved under multi-rollout evaluation (Fig. 2(b)): hy–ko slightly decreases (+5.17 → +4.61), lo–th is preserved (+2.81→ +3.18), and eu–ko also maintains a pos- itive effect (+6.32 → +2.78). This consistency suggests that these effects are not artifacts of a spe- cific decoding setting, but directionally consistent patterns that recur across certain language combina- tions (statistical significance and cross-benchmark checks are reported in Appendices J and K). Case Study: Maltese. Another notable case is Maltese (mt). In Fig. 2(a), Maltese shows large pos- itive effects with Russian (+8.62), Thai (+6.90), and Spanish (+6.61) skeletons, and all three pairs satisfy nominal significance under McNemar’s test (Appendix J). However, these effects are attenu- ated or even reversed under multi-rollout evalu- ation (e.g., mt–ru:−0.23). Therefore, unlike the directionally consistent cases discussed above, the positive effects observed for Maltese are sensitive to the evaluation protocol and to the benchmark (Appendix K), and we do not treat them as evi- dence of a generalizable positive effect. Since the results in this section alone are insufficient to deter- mine the source of this pattern, we further examine it in Sec. 5. Asymmetric Negative EffectsSurface-level sim- ilarities such as shared scripts or geographic prox- imity do not necessarily lead to positive transfer. Mongolian (mn) and Russian share the Cyrillic script, but differ in word order and morphological structure: Mongolian is an agglutinative language, whereas Russian is fusional. Accordingly, Russian skeletons lead to performance degradation under both greedy decoding (−5.00) and multi-rollout evaluation (−2.65). Similarly, Uyghur (ug) and Chinese are geographically proximate but struc- turally different: Uyghur is an SOV agglutinative language, whereas Chinese is an SVO analytic lan- guage. Chinese skeletons degrade Uyghur perfor- mance (−6.45greedy;−1.12multi-rollout). These cases suggest that surface-level similarity alone is insufficient to predict the direction of skeleton transfer; this negative pattern largely persists on MSVAMP (Appendix K). The Necessity of LASEF. Taken together, the effects of skeleton language fall into three pat- terns: (1) directionally consistent effects, where the sign of positive transfer holds under both greedy and multi-rollout evaluation; (2) evaluation- and benchmark-dependent effects, where large greedy gains are attenuated under multi-rollout evaluation or on an unseen benchmark; and (3) asymmetric negative effects, where negative transfer occurs de- spite shared surface affinity. These heterogeneous, largely exploratory patterns (Appendices J and K) show that a single evaluation protocol is insufficient to fully characterize skeleton-language effects, sup- porting the need for LASEF as a multi-level empir- ical analysis framework rather than a rule-based selector. 5 Analysis 1050510 LLM-Generated Skeleton (%) 10 5 0 5 10 Translated Skeleton (quality-controlled) (%) UG-ZH MT-ES MN-RU MT-RU EU-KO HY-KO MT-KO LO-TH MT-TH Effect persists (surface form) Negative in both conditions Translation Ablation: Disentangling Quality from Surface Form Other pairs (n=86) Stable cross-lingual Maltese (eval-dependent) Asymmetric negative Figure 3: Translation ablation results. Each point is a (target, skeleton) pair, with∆from LLM-generated skeletons on the x-axis and translated skeletons on the y-axis. Points near the diagonal indicate effects robust to skeleton quality. The effects observed in Sec. 4.5 may conflate two factors: differences in skeleton generation qual- ity and cross-lingual alignment between skeleton and target language. To disentangle these factors, we conduct a translation ablation. Specifically, we translate the English skeleton into five non-English languages (zh, es, ru, ko, th) using GPT-5-mini, and then run greedy-decoding inference with the translated skeletons, comparing them against the English-skeleton baseline. Since the semantic content is fixed, any observed difference more directly reflects surface-form vari- ation rather than skeleton generation quality. Fig. 3 visualizes the results: thex-axis denotes∆with LLM-generated skeletons, they-axis denotes∆ with translated skeletons, and points near the diag- onal (y=x) indicate effects that are less sensitive to skeleton generation quality. The qualitative LLM- as-a-Judge analysis is provided in Appendix F. Directionally Consistent Pairs Persist The di- rectionally consistent pairs from Sec. 4.5 fall in the upper-right region of Fig. 3 and retain a positive sign under translation ablation: Basque–Korean (+6.32 → +4.42), Armenian–Korean (+5.17 → +1.61), and Lao–Thai (+2.81 → +0.40). Al- though the magnitude of∆decreases, the positive sign is preserved, suggesting that these effects are not solely attributable to skeleton generation qual- ity. Maltese: A Fragile Surface Effect Maltese is the clearest evaluation-dependent case: large greedy gains with several non-English skeletons are attenuated or reversed under multi-rollout eval- uation. The translation ablation shows this is not a generation-quality artifact: with content fixed and only the surface language varied, the gains stay positive (mt–ru:+5.15, mt–ko:+5.06). It is thus a genuine surface-form effect, but one sensitive to the decoding protocol and to the evaluation bench- mark: on MSVAMP, the mt–ru gain reverses (Ap- pendix K). Asymmetric Negatives Persist In contrast, the asymmetric negative pairs, Mongolian–Russian (−3.43) and Uyghur–Chinese (−6.94), lie near the diagonal in the lower-left quadrant of Fig. 3: both LLM-generated and translated skeletons yield comparable negative transfer, so it is not a quality artifact. These pairs share surface affinity (script or geography) but lack structural alignment (Ap- pendix D), so the effect tracks misalignment, not quality or surface similarity. Together with the stable and Maltese pairs, this shows that skeleton-language effects are not re- ducible to generation quality but reflect genuine skeleton–target interactions—effects that cannot be predicted a priori and must be found empirically, the role of LASEF. 6 Conclusion We analyzed the problem of skeleton language selection in multilingual mathematical reasoning through the Language-Aware Skeleton Exploration Framework (LASEF), an empirical analysis frame- work rather than an automatic selector of an opti- mal skeleton language. Our experiments show that English skeletons are a reasonable default, but their average gains are small and they are not a one-size- fits-all solution. By combining greedy decoding, multi-rollout evaluation, translation ablation, and cross-benchmark validation, we further find that skeleton-language effects can be categorized into three patterns: directionally consistent, evaluation- and benchmark-dependent, and asymmetric neg- ative. These results answer our research question conditionally rather than universally: skeleton lan- guage is a context-dependent design variable that cannot be fully explained by generation quality alone, and effective—or risky—language combi- nations should be identified through multi-level exploration. Limitations Our study has several limitations. First, we focused exclusively on mathematical reasoning; generaliz- ing findings to domains like commonsense reason- ing or code generation requires further validation. Second, our training-free approach ensures broad applicability but precludes potential gains from fine-tuning. Third, we maintained a fixed skele- ton structure to isolate language effects, leaving the interplay between skeleton design and language choice unexplored. Finally, while we covered diverse languages, the full LASEFconfiguration search remains computationally expensive and is intended as an offline diagnostic rather than a per- deployment step; results from an existing bench- mark can help flag risky skeleton candidates but should be revalidated whenever the benchmark, model, or task changes (Appendix K). Developing efficient predictive methods for this search remains future work. Acknowledgments This work was supported by Institute of Informa- tion & Communications Technology Planning & Evaluation (IITP) grants funded by the Korea gov- ernment (MSIT) (No. 2022-0-00871, Development of AI Autonomy and Knowledge Enhancement for AI Agent Collaboration; No. RS-2026-25525363, AEGIS: Agentic Experts for Generative-AI Inspec- tion Solution). This work also utilized GPU re- sources from the “Advanced GPU Utilization Sup- port Program” funded by MSIT, Republic of Korea (awarded to KyungTae Lim). References Kabir Ahuja, Harshita Diddee, Rishav Hada, Milli- cent Ochieng, Krithika Ramesh, Prachi Jain, Ak- shay Nambi, Tanuja Ganu, Sameer Segal, Mohamed Ahmed, and 1 others. 2023. Mega: Multilingual eval- uation of generative ai. In Proceedings of the 2023 Conference on Empirical Methods in Natural Lan- guage Processing, pages 4232–4267. Linzheng Chai, Jian Yang, Tao Sun, Hongcheng Guo, Jiaheng Liu, Bing Wang, Xinnian Liang, Jiaqi Bai, Tongliang Li, Qiyao Peng, and 1 others. 2025. xcot: Cross-lingual instruction tuning for cross-lingual chain-of-thought reasoning.In Proceedings of the AAAI Conference on Artificial Intelligence, vol- ume 39, pages 23550–23558. Nuo Chen, Zinan Zheng, Ning Wu, Ming Gong, Dong- mei Zhang, and Jia Li. 2024. Breaking language barriers in multilingual mathematical reasoning: In- sights and observations. In Findings of the Associ- ation for Computational Linguistics: EMNLP 2024, pages 7001–7016. Wenhu Chen, Xueguang Ma, Xinyi Wang, and William W Cohen. 2022.Program of thoughts prompting: Disentangling computation from reason- ing for numerical reasoning tasks. Transactions on Machine Learning Research. Karl Cobbe, Vineet Kosaraju, Mohammad Bavarian, Mark Chen, Heewoo Jun, Lukasz Kaiser, Matthias Plappert, Jerry Tworek, Jacob Hilton, Reiichiro Nakano, and 1 others. 2021.Training verifiers to solve math word problems.arXiv preprint arXiv:2110.14168. Julen Etxaniz, Gorka Azkune, Aitor Soroa, Oier Lopez de Lacalle, and Mikel Artetxe. 2024. Do multilingual language models think better in english? In Proceed- ings of the 2024 Conference of the North American Chapter of the Association for Computational Lin- guistics: Human Language Technologies (Volume 2: Short Papers), pages 550–564. Aaron Grattafiori, Abhimanyu Dubey, Abhinav Jauhri, Abhinav Pandey, Abhishek Kadian, Ahmad Al-Dahle, Aiesha Letman, Akhil Mathur, Alan Schelten, Alex Vaughan, Amy Yang, Angela Fan, Anirudh Goyal, Anthony Hartshorn, Aobo Yang, Archi Mitra, Archie Sravankumar, Artem Korenev, Arthur Hinsvark, and 542 others. 2024. The llama 3 herd of models. Preprint, arXiv:2407.21783. Dan Hendrycks, Collin Burns, Saurav Kadavath, Akul Arora, Steven Basart, Eric Tang, Dawn Song, and Jacob Steinhardt. 2021. Measuring mathematical problem solving with the math dataset. In Thirty- fifth Conference on Neural Information Processing Systems Datasets and Benchmarks Track (Round 2). Haoyang Huang, Tianyi Tang, Dongdong Zhang, Wayne Xin Zhao, Ting Song, Yan Xia, and Furu Wei. 2023. Not all languages are created equal in llms: Improving multilingual capability by cross-lingual- thought prompting. In Findings of the Association for Computational Linguistics: EMNLP 2023, pages 12365–12394. Armand Joulin, Edouard Grave, Piotr Bojanowski, and Tomas Mikolov. 2017. Bag of tricks for efficient text classification. In Proceedings of the 15th Conference of the European Chapter of the Association for Com- putational Linguistics: Volume 2, Short Papers, pages 427–431. Association for Computational Linguistics. Tushar Khot, Harsh Trivedi, Matthew Finlayson, Yao Fu, Kyle Richardson, Peter Clark, and Ashish Sab- harwal. 2022. Decomposed prompting: A modular approach for solving complex tasks. In The Eleventh International Conference on Learning Representa- tions. Takeshi Kojima, Shixiang Shane Gu, Machel Reid, Yu- taka Matsuo, and Yusuke Iwasawa. 2022. Large lan- guage models are zero-shot reasoners. Advances in neural information processing systems, 35:22199– 22213. Hynek Kydlí ˇ cek. Math-Verify: Math Verification Li- brary. Huiyuan Lai and Malvina Nissim. 2024. mcot: Multi- lingual instruction tuning for reasoning consistency in language models. In Proceedings of the 62nd An- nual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pages 12012– 12026. Nahyun Lee, Yeongseo Woo, Hyunwoo Ko, and Gui- jin Son. 2025. Controlling language confusion in multilingual llms. In ACL 2025 Student Research Workshop. Bryan Li, Tamer Alkhouli, Daniele Bonadiman, Niko- laos Pappas, and Saab Mansour. 2024. Eliciting better multilingual structured reasoning from llms through code. In Proceedings of the 62nd Annual Meeting of the Association for Computational Lin- guistics (Volume 1: Long Papers), pages 5154–5169. Hunter Lightman, Vineet Kosaraju, Yuri Burda, Harri- son Edwards, Bowen Baker, Teddy Lee, Jan Leike, John Schulman, Ilya Sutskever, and Karl Cobbe. 2023. Let’s verify step by step. In The Twelfth Inter- national Conference on Learning Representations. Alexander H. Liu, Kartik Khandelwal, Sandeep Subramanian, Victor Jouault, Abhinav Rastogi, Adrien Sadé, Alan Jeffares, Albert Jiang, Alexan- dre Cahill, Alexandre Gavaudan, Alexandre Sablay- rolles, Amélie Héliou, Amos You, Andy Ehrenberg, Andy Lo, Anton Eliseev, Antonia Calvi, Avinash Sooriyarachchi, Baptiste Bout, and 101 others. 2026. Ministral 3. Preprint, arXiv:2601.08584. Chaoqun Liu, Wenxuan Zhang, Yiran Zhao, Luu Anh Tuan, and Lidong Bing. 2025. Is translation all you need? a study on solving multilingual tasks with large language models. In Proceedings of the 2025 Con- ference of the Nations of the Americas Chapter of the Association for Computational Linguistics: Human Language Technologies (Volume 1: Long Papers), pages 9594–9614. Xuefei Ning, Zinan Lin, Zixuan Zhou, Zifu Wang, Huazhong Yang, and Yu Wang. 2023. Skeleton-of- thought: Large language models can do parallel de- coding. Proceedings ENLSP-I. OpenAI, Josh Achiam, Steven Adler, Sandhini Agarwal, Lama Ahmad, Ilge Akkaya, Florencia Leoni Ale- man, Diogo Almeida, Janko Altenschmidt, Sam Alt- man, Shyamal Anadkat, Red Avila, Igor Babuschkin, Suchir Balaji, Valerie Balcom, Paul Baltescu, Haim- ing Bao, Mohammad Bavarian, Jeff Belgum, and 262 others. 2024. Gpt-4 technical report. Preprint, arXiv:2303.08774. Jirui Qi, Shan Chen, Zidi Xiong, Raquel Fernández, Danielle Bitterman, and Arianna Bisazza. 2025a. When models reason in your language: Controlling thinking language comes at the cost of accuracy. In Findings of the Association for Computational Lin- guistics: EMNLP 2025, pages 20279–20296. Jirui Qi, Shan Chen, Zidi Xiong, Raquel Fernández, Danielle S Bitterman, and Arianna Bisazza. 2025b. When models reason in your language: Controlling thinking language comes at the cost of accuracy. In Findings of the Association for Computational Lin- guistics: EMNLP 2025, pages 20279–20296. Rui Qi, Zhibo Man, Yufeng Chen, Fengran Mo, Jinan Xu, and Kaiyu Huang. 2025c. Sot: Structured-of- thought prompting guides multilingual reasoning in large language models. In Findings of the Associa- tion for Computational Linguistics: EMNLP 2025, pages 11024–11039. Shenbin Qian, Archchana Sindhujan, Minnie Kabra, Diptesh Kanojia, Constantin Orasan, Tharindu Ranas- inghe, and Fred Blain. 2024. What do large lan- guage models need for machine translation evalu- ation? In Proceedings of the 2024 Conference on Empirical Methods in Natural Language Processing, pages 3660–3674. Libo Qin, Qiguang Chen, Fuxuan Wei, Shijue Huang, and Wanxiang Che. 2023. Cross-lingual prompt- ing: Improving zero-shot chain-of-thought reasoning across languages. In Proceedings of the 2023 Con- ference on Empirical Methods in Natural Language Processing, pages 2695–2709. Qwen, :, An Yang, Baosong Yang, Beichen Zhang, Binyuan Hui, Bo Zheng, Bowen Yu, Chengyuan Li, Dayiheng Liu, Fei Huang, Haoran Wei, Huan Lin, Jian Yang, Jianhong Tu, Jianwei Zhang, Jianxin Yang, Jiaxi Yang, Jingren Zhou, and 25 oth- ers. 2025.Qwen2.5 technical report.Preprint, arXiv:2412.15115. Freda Shi, Mirac Suzgun, Markus Freitag, Xuezhi Wang, Suraj Srivats, Soroush Vosoughi, Hyung Won Chung, Yi Tay, Sebastian Ruder, Denny Zhou, and 1 others. 2022. Language models are multilingual chain-of- thought reasoners. In The Eleventh International Conference on Learning Representations. Gemini Team, Rohan Anil, Sebastian Borgeaud, Jean- Baptiste Alayrac, Jiahui Yu, Radu Soricut, Johan Schalkwyk, Andrew M. Dai, Anja Hauth, Katie Millican, David Silver, Melvin Johnson, Ioannis Antonoglou, Julian Schrittwieser, Amelia Glaese, Jilin Chen, Emily Pitler, Timothy Lillicrap, Angeliki Lazaridou, and 1332 others. 2025. Gemini: A fam- ily of highly capable multimodal models. Preprint, arXiv:2312.11805. Bin Wang, Zhengyuan Liu, Xin Huang, Fangkai Jiao, Yang Ding, AiTi Aw, and Nancy Chen. 2024. Seae- val for multilingual foundation models: From cross- lingual alignment to cultural reasoning. In Proceed- ings of the 2024 Conference of the North American Chapter of the Association for Computational Lin- guistics: Human Language Technologies (Volume 1: Long Papers), pages 370–390. Lei Wang, Wanyu Xu, Yihuai Lan, Zhiqiang Hu, Yunshi Lan, Roy Ka-Wei Lee, and Ee-Peng Lim. 2023. Plan- and-solve prompting: Improving zero-shot chain-of- thought reasoning by large language models. In Pro- ceedings of the 61st Annual Meeting of the Associa- tion for Computational Linguistics (Volume 1: Long Papers), pages 2609–2634. Mingyang Wang, Lukas Lange, Heike Adel, Yunpu Ma, Jannik Strötgen, and Hinrich Schuetze. 2025a. Lan- guage mixing in reasoning language models: Pat- terns, impact, and internal causes. In Proceedings of the 2025 Conference on Empirical Methods in Natu- ral Language Processing, pages 2637–2665, Suzhou, China. Association for Computational Linguistics. Qihan Wang, Shidong Pan, Tal Linzen, and Emily Black. 2025b. Multilingual prompting for improving LLM generation diversity. In Proceedings of the 2025 Conference on Empirical Methods in Natural Lan- guage Processing, pages 6378–6400, Suzhou, China. Association for Computational Linguistics. Yiming Wang, Pei Zhang, Jialong Tang, Haoran Wei, Baosong Yang, Rui Wang, Chenshu Sun, Feitong Sun, Jiran Zhang, Junxuan Wu, Qiqian Cang, Yichang Zhang, Fei Huang, Junyang Lin, Fei Huang, and Jin- gren Zhou. 2025c. Polymath: Evaluating mathemat- ical reasoning in multilingual contexts. The Thirty- Ninth Annual Conference on Neural Information Pro- cessing Systems. Jason Wei, Xuezhi Wang, Dale Schuurmans, Maarten Bosma, Fei Xia, Ed Chi, Quoc V Le, Denny Zhou, and 1 others. 2022. Chain-of-thought prompting elic- its reasoning in large language models. Advances in neural information processing systems, 35:24824– 24837. Shunyu Yao, Dian Yu, Jeffrey Zhao, Izhak Shafran, Tom Griffiths, Yuan Cao, and Karthik Narasimhan. 2023. Tree of thoughts: Deliberate problem solving with large language models. Advances in neural informa- tion processing systems, 36:11809–11822. Denny Zhou, Nathanael Schärli, Le Hou, Jason Wei, Nathan Scales, Xuezhi Wang, Dale Schuurmans, Claire Cui, Olivier Bousquet, Quoc V Le, and 1 oth- ers. 2022. Least-to-most prompting enables complex reasoning in large language models. In The Eleventh International Conference on Learning Representa- tions. Wenhao Zhu, Shujian Huang, Fei Yuan, Shuaijie She, Jiajun Chen, and Alexandra Birch. 2024. Question translation training for better multilingual reasoning. In Findings of the Association for Computational Linguistics ACL 2024, pages 8411–8423. A Skeleton A.1 Skeleton Generation Method The skeleton generation pipeline proposed in this study is constructed as follows. Input Configuration When generating a skele- ton, the model is provided with the problem text together with three few-shot examples. These few- shot examples constrain the output style and ab- straction level of the skeleton, ensuring that the skeleton is produced in the specified output lan- guageℓ s . Meanwhile, the input question itself is presented in the target languageℓ t , and the model generates the skeleton based on this input. Two ComponentsInstead of directly presenting the computational steps or explicit reasoning pro- cedures required to solve the problem, the skeleton is designed to abstract and represent the underlying structural elements and the conceptual problem- solving framework. The generated skeleton is orga- nized into the following two core components: 1.Problem Structure: Identifies and defines the Objects, Variables, Relationships, Con- straints, and Problem Nature included in the problem. 2.Key Concepts / Tools: Describes the mathe- matical/scientific principles and tools essen- tial for solving the problem using standardized academic terminology. Generation Constraints To maintain the level of abstraction and consistency of the skeleton, the generation process is subject to the following con- straints. First, the final answer or any specific nu- merical values are not mentioned explicitly, and the numbers or formulas appearing in the reason- ing path are replaced with their conceptual roles (e.g., “coefficient”, “boundary value”, “ratio”). Sec- ond, instead of computation-centered verbs such as “calculate” or “compute” the model is encour- aged to use verbs that induce conceptual reasoning, such as “identify”, “express”, and “relate”. Third, while the essential logical structure of the problem- solving process is preserved, peripheral procedu- ral details, redundant descriptions, and errors con- tained in the reasoning path are excluded. Through these constraints, the skeleton is guided to extract the structural essence of the problem without di- rectly presenting the solution method. A.2 System Prompt for Skeleton Generation The details of the System Prompt applied to the model for skeleton generation are presented in Tab. 3. This prompt is designed to ensure that the model does not directly present the final answer or explicit solution procedures (How), but instead abstracts the structural characteristics and core con- cepts (What) of the problem to construct a concep- tual problem-solving plan. The prompt used in the subsequent stage, where the model solves the prob- lem based on the generated skeleton, is presented as follows. Solver Prompt for Skeleton Question: [Problem Text] Reasoning Skeleton: [Generated Skeleton] Instructions: 1. Follow the Reasoning Skeleton above. 2. Verify numbers against the Question text. 3. Let’s think step-by-step. Respond in [Target Language]. Final Answer Format: answer A.3 Few-shot Examples Used for Skeleton Reasoning Skeleton Few-shot Example 1 (Korean) Question:삼각형에 길이가5cm와12cm인 두 변이 있으며 ,그사이에 직각이 있습니다.세 번째 변의 길이는 얼마입니까? (A triangle has sides of length 5 cm and 12 cm, with a right angle between them. What is the length of the third side?) Skeleton: Problem Structure - Geometric figure: right triangle with three sides - Known quantities: two perpendicular sides (5 cm, 12 cm), right angle constraint - Unknown: third side (hypotenuse) connecting the two legs - Relationship: side lengths satisfy distance relation en- forced by right angle property Key Concepts / Tools - Pythagorean theorem - Right triangle geometry System prompt for skeleton generation Your Role: You generate minimal, high-level reasoning skeletons for math and science problems. Language Requirement: Always respond in English, regardless of the language used in the user query. Goal: Provide only the *essential structural characteristics* of the problem. You must describe WHAT the problem structure is, never HOW to solve it. Output Format: Provide exactly two sections below. 1. **Problem Structure** Identify and characterize: - Objects & Variables: Types of entities (quantities, sets, functions, geometric figures), separating constants/parameters from variables. Distinguish knowns vs. unknowns. - Relationships: The nature of connections between entities (e.g., proportional, inverse, functional dependency, spatial). Describe the *type* of relationship, not the derivation. - Constraints: Conditions defining the problem space (equations, inequalities, domain restrictions, boundary conditions, conservation laws). - Problem Nature: (Only if relevant) Briefly note the structural type (e.g., Optimization, Combinatorial, Inverse problem). Constraints for this section: - Must be problem-specific, not generic. - Must NOT include solution steps, strategies, formulas, or numerical manipulations. 2. Key Concepts / Tools List relevant mathematical or scientific principles using standard terminology (e.g., “Pythagorean Theorem”, “Conservation of Energy”, “Bayes’ Theorem”). *Constraints for this section:* - Use established concept names only. - Do NOT explain how to apply them or list formulas. Table 3: The prompt instructs the model to produce a concise, high-level outline of the reasoning process, while omitting detailed computations and final answers. Skeleton Few-shot Example 2 (Korean) Question: 5명의 후보(Alice, Bob, Charlie, Dave, Eve)중에서2명의 위원을 선출해야 합니다.그러나 Alice와Bob은함께위원이될수없습니다.위원회를 구성하는 방법은몇 가지입니까 ? (A committee needs to pick 2 members from a group of 5 candidates. However, Alice and Bob cannot be on the committee together. How many ways can the committee be formed?) Skeleton: Problem Structure - Discrete selection problem: choosing k=2 elements from n=5 candidates - Known: total candidates (5), committee size (2), pair- wise exclusion constraint - Unknown: number of valid selection configurations - Constraint: pairwise exclusion (Alice and Bob cannot co-occur) Key Concepts / Tools - Combinatorial counting (combinations) - Constraint satisfaction - Complementary counting Few-shot Example 3: Rate Problem Question:기계A는8시간동안 시간당100개를 생산합니다 .기계B는4시간 가동 후1시간휴식하고, 그 후3시간동안 시간당120개를생산합니다.총 생산량은 얼마입니까? (Machine A runs for 8 hours producing 100 units/hour. Machine B runs for 4 hours, rests for 1 hour, then runs for 3 hours at 120 units/hour. What is the total production?) Skeleton: Problem Structure - Two independent production processes with different temporal patterns - Variables: production rates, time durations, total output - Machine A: single continuous active interval (8 hours @ 100 units/hour) - Machine B: two disjoint active intervals separated by idle period - Relationship: total output = sum of (rate×active dura- tion) across all segments Key Concepts / Tools - Rate-time-quantity relationships - Piecewise continuous processes - Linear aggregation For each target language (Korean, Spanish, Chi- nese, Thai, Swahili), we used few-shot examples where the questions were translated into the respec- tive language, while the skeleton responses were generated in English in all cases. A.4 Example of Skeleton Generation Example (1): Skeleton Generation Question:한 자전거가분당20피트의 속도로 이동하 고 있다.이 자전거의 속도를 초당 인치로표현하면 얼마인가 ? (A bicycle is traveling at 20 feet per minute. What is the bicycle’s speed expressed in inches per second?) Skeleton: Problem Structure - Object: bicycle - Known: speed in feet per minute (20 ft/min) - Unknown: speed in inches per second - Relationships: conversion between units of length (feet to inches) and time (minutes to seconds) - Constraints: 1 foot = 12 inches, 1 minute = 60 seconds Key Concepts / Tools - Unit conversion - Dimensional analysis Example (2): Skeleton Generation Question: 6명의마녀와10명의마법사가비밀모임에 참석하였다 .마녀들은 각마법사와 악수를 하기로 결정했다.마녀들끼리는 이미모두친한친구이므로 서로 악수를 하지 않고,마법사들끼리는 서로 싫어 해서 악수를 하지 않았다.이모임에서 이루어진 총 악수의수는몇인가? (Six witches and ten sorcerers are at an arcane mixer. The witches have decided to shake hands with each of the sorcerers. None of the witches bothered to shake each other’s hands, since they are all good friends already, and the sorcerers all sort of hate each other and did not shake hands with other sorcerers. How many handshakes took place at the mixer?) Skeleton: Problem Structure - Discrete interaction problem: handshakes between two distinct groups - Known: number of witches (6), number of wizards (10) - Unknown: total number of handshakes - Relationship: each witch shakes hands with each wizard exactly once - Constraints: no handshakes between witches, no hand- shakes between wizards Key Concepts / Tools - Combinatorial counting (specifically, Cartesian product of two sets) - Bipartite graph theory (each handshake represents an edge between two disjoint sets) B Evaluation Setup B.1 Chain-of-Thought Prompt We adopt Chain-of-Thought (Wei et al., 2022) as our baseline approach. The prompt is shown below. Prompt for CoT System: You are a math problem solver. Solve the given problem. At the end, clearly state the final answer in the following format: final answer User: Question: [Problem Text] Let’s think step by step. Respond in [Target Language]. B.2 Evaluation Protocol Decoding Strategy To minimize variability in the reasoning process across all experiments, Greedy Decoding was used. The temperature was set to 0, selecting the token with the highest proba- bility at each generation step. Exact Match (EM) Evaluation Quantitative evaluation was performed using themath-verify library (Kydlí ˇ cek). The Exact Match (EM) score was calculated based on whether the answer ex- tracted in the ‘ ’ format from the model’s final output matched the Ground Truth. Language Consistency Filtering To ensure a fair comparison of reasoning capabilities, we in- cluded in the evaluation only those samples whose generated output language matched the target lan- guage. In particular, we filtered samples satisfying the conditionℓ a = ℓ t , thereby preventing evalua- tion bias that may arise from code-switching or un- intended language mixing. Language identification for the generated responses was performed using the FastText classifier (Joulin et al., 2017). Apply- ing this filter to the high-resource setting of Tab. 1 retains 98.8/98.4% (MGSM), 93.1/93.4% (MATH- 500), and 92.5/92.4% (PolyMath) of samples for target-language/English-translated queries; reten- tion varies more substantially across low-resource languages, and per-language sample counts to- gether with unfiltered accuracies are reported in Appendix L. In addition, to ensure a fair comparison between the baseline and the skeleton-based method (e.g., CoT-ℓ t vs. +SKELETON), evaluation was con- ducted only on samples that satisfied the condition ℓ q = ℓ t in both settings. This filtering ensured that the reasoning performance was compared under identical input–output language conditions. Intersection-based Pairing. For comparisons across skeleton languages in Sec. 4.5, we adopt intersection-based pairing to ensure fair compari- son. Specifically, each (target language, skeleton language) pair is compared against its English- skeleton counterpart only on the subset of items where both settings produceℓ a = ℓ t . To ensure reliable analysis, we further require each target lan- guage to retain at least 180 valid samples on aver- age across the five non-English skeleton languages (zh, es, ru, ko, th); target languages with fewer sam- ples are excluded. In total, MGSM-LowResource covers 34 target languages; the 19 that satisfy this criterion form the main analysis set of Sec. 4.5, and combined with the five non-English skeleton languages they yield the 95 target–skeleton pairs tested in Appendix J. Language Forcing Following the approach pro- posed by Qi et al. (2025a), we incorporate the in- struction “Respond in l t ” along with language- specific trigger phrases. Specifically, we use trig- gers such as “좋습니다,” for Korean, “Sawa,” for Swahili, and “Bien,” for Spanish to encourage the model to initiate generation in the target language. B.3 Mathematical Benchmarks In this experiment, to comprehensively evaluate the mathematical reasoning capabilities of multilingual LLMs, we utilized three benchmarks with differing problem difficulties and linguistic characteristics. MGSM (Multilingual Grade School Math) A multilingual benchmark built upon the 250-sample test set of GSM8K (Cobbe et al., 2021) (Shi et al., 2022), consisting of elementary-level arith- metic problems. It is widely used as a standard metric for measuring basic arithmetic reasoning skills in multilingual environments. In addition to the existing 10 languages (Bengali, Chinese, En- glish, French, German, Japanese, Russian, Spanish, Swahili, Thai), we expanded the evaluation scope to include Korean 2 and various low-resource lan- guages. MATH-500A subset of 500 representative ques- tions selected by topic and difficulty from the high- difficulty mathematics dataset MATH (Hendrycks et al., 2021). It covers advanced mathematical fields such as Algebra, Geometry, and Calculus, requir- ing complex mathematical thinking and problem- 2 https://huggingface.co/datasets/freewheelin/ mgsm_ko solving skills beyond simple calculations. Since MATH is originally an English-only benchmark, we utilized existing high-quality human transla- tions 3 for our multilingual evaluation. PolyMath A recently proposed high-difficulty multilingual math reasoning benchmark (Wang et al., 2025c), containing 500 complex problems (125 per difficulty level) across four difficulty levels in 18 languages. Composed of high-quality ques- tions reflecting the cultural and linguistic contexts of each language region rather than simple transla- tions, it is suitable for analyzing the deep multilin- gual reasoning limitations and performance gaps of LLMs. B.4 Models We selected QWEN2.5 and LLAMA 3.1, the latest open-source model families with verified multilin- gual processing capabilities and reasoning perfor- mance. Qwen2.5(Qwen et al., 2025) Developed by Al- ibaba Cloud, the QWEN2.5 series has demonstrated exceptional performance, particularly in mathemat- ical reasoning and coding. It offers various param- eter sizes ranging from 0.5B to 72B and is pre- trained on a vast multilingual corpus, exhibiting strong performance even in non-English languages. In this study, we used the 7B, 14B, and 72B mod- els to analyze the effect of skeletons according to model scale expansion. Llama 3.1(Grattafiori et al., 2024) As the suc- cessor to Meta’s LLAMA 3, LLAMA 3.1 features an extended context window of 128K tokens and en- hanced multilingual support. To verify whether the skeleton-based methodology is effective in general- purpose LLMs not specialized for specific domains, we used the 8B and 70B models in our experiments. C Details of MGSM Low-Resource Dataset Construction This section details the construction process and rigorous quality control system of the MGSM Low- Resource benchmark newly established in this study. To expand the existing MGSM dataset to 3 https://huggingface.co/datasets/ appier-ai-research/Multilingual-MATH-500, https://huggingface.co/datasets/davidkim205/ ko-math-500,https://huggingface.co/datasets/ iapp/math-500-th 34 low-resource languages, we introduced a Dual- Model Translation Pipeline applying prompt engi- neering and a multi-stage verification procedure. C.1 Selection Criteria for Low-Resource Languages Target languages were selected from ultra-low- resource languages that account for less than 0.05%of the data within the large-scale corpus allenai/c4 4 . This was to identify languages that are not sufficiently included in the pre-training data of modern LLMs, leading to underestimated per- formance or lack of research. The 34 finally selected languages were chosen considering typological diversity, including various language families, writing systems, and geograph- ical distributions. The detailed list is provided in Tab. 4. RegionLanguagesScript AfricaAmharicGe’ez EuropeBasque, Maltese, ArmenianLatin, Armenian Central AsiaKazakh, Kyrgyz, Uyghur, PashtoCyrillic, Arabic South AsiaNepali, Sinhala, Sindhi, Punjabi, Gu- jarati, Marathi, Odia, Tamil, Malay- alam Indic, Arabic SE Asia & PacificLao, MongolianLao, Cyrillic Table 4: List of the 19 low-resource target languages included in the MGSM-LowResource dataset. C.2 Dataset Construction Pipeline Thedatasetconstructionadopteda dual-modelpipelineutilizingGem- ini (gemini-3.0-flash) (Team et al., 2025) and GPT (gpt-5-mini) (OpenAI et al., 2024) in parallel to eliminate single-model dependency and maximize translation quality. The entire process consists of three stages: Generation-Validation- Selection. C.2.1 Stage 1: Prompt-Based Generation and Numerical Consistency Check For each problem, two models independently gener- ate translation candidates (T). To prevent distortion of mathematical symbols or unnecessary additions and ensure translation accuracy, we designed and applied a system prompt including a professional translator persona and four key constraints (numeri- cal preservation, meaning retention, natural fluency, and output format compliance). The details of the prompt used are as follows: 4 https://huggingface.co/datasets/allenai/c4 Translation System Prompt "You are a professional translator. Translate the given text to [Target Language]. Rules: 1. Preserve all numbers, mathematical expressions, and proper nouns exactly as they appear 2. Maintain the original meaning and context 3. Use natural, fluent [Target Language] 4. Only output the translation, nothing else" Particularly, to guarantee numerical accuracy, which is core information in math word problems, a Numerical Preservation check was performed immediately after generation. We calculated the recall of numerical entities between the source text (E) and the translated text (T), and if it fell below the threshold (80%), the translation was immedi- ately discarded, and the regeneration process was triggered. Recall num (T,E) < 0.8 =⇒ Regenerate(1) Through this strict filtering process, numerical errors were preemptively blocked, and only valid candidates adhering to prompt constraints were passed to the next stage. C.2.2 Stage 2: Back-Translation Quality Selection (BTQS) To select the optimal translation among the can- didates (T gemini ,T gpt ) that passed the numerical check, we applied the Back-Translation Quality Selection (BTQS) methodology. 1.Back-Translation: Each candidate translation is back-translated into English (E ′ ) using the model used for its generation. 2.Semantic Similarity Measurement: To eval- uate semantic equivalence between the source (E) and back-translated text (E ′ ), cosine sim- ilarity is calculated using theBAAI/bge-m3 encoder. s(T ) = cos φ(E),φ(E ′ ) (2) 3. Final Selection: The translation with the higher similarity score is adopted as the final data. T ∗ =arg max T∈T gemini ,T gpt s(T )(3) Target LangAcc (%)∆Target LangAcc (%)∆ English (Baseline)90.00–Quechua (qu)89.60 -0.40 Tamil (ta)93.60 +3.60Guarani (gn)90.000.00 Gujarati (gu)91.60 +1.60 Basque (eu)90.000.00 Kannada (kn)91.20 +1.20Amharic (am)90.000.00 Khmer (km)91.20 +1.20Javanese (jv)90.000.00 Sinhala (si)91.20 +1.20Uyghur (ug)89.60 -0.40 Uzbek (uz)91.20 +1.20 Tajik (tg)89.60 -0.40 Yoruba (yo)90.80 +0.80Sundanese (su)89.60 -0.40 Kyrgyz (ky)90.80 +0.80Burmese (my)89.60 -0.40 Sindhi (sd)90.40 +0.40Georgian (ka)89.60 -0.40 Nepali (ne)90.40 +0.40 Pashto (ps)89.60 -0.40 Punjabi (pa)90.40 +0.40Armenian (hy)89.60 -0.40 Maltese (mt)90.40 +0.40Malayalam (ml)89.60 -0.40 Lao (lo)90.40 +0.40 Malagasy (mg)89.60 -0.40 Somali (so)90.000.00Mongolian (mn)89.20 -0.80 Odia (or)90.000.00 Kazakh (k)89.20 -0.80 Marathi (mr)90.000.00Kurdish (ku)89.20 -0.80 Cebuano (ceb)90.000.00 Table 5: Reasoning accuracy of Qwen2.5-7B on MGSM tasks using Round-trip Translation (English→Pivot →English). We used the target languages as pivots to validate the semantic consistency of the translated dataset.∆denotes the performance difference relative to the English baseline. C.3 Quality Control and Verification Results To confirm the reliability of the constructed dataset, we doubly verified linguistic accuracy and logi- cal integrity based on the outputs of the BTQS pipeline. First, samples with BTQS similarity scores be- low the threshold (0.8) from the previous stage were considered defective and primarily excluded. Instead of simple exclusion, a Regeneration loop was immediately executed for these samples to it- eratively correct them until all data met the quality standards. For the finally selected samples, Round-trip Reasoning Validation was performed to check the preservation of mathematical essence. To maxi- mize efficiency, this was conducted by reusing the back-translated text generated during the BTQS evaluation process without additional translation. That is, English text generated via 34 target lan- guages as pivots was input into the reasoning model (QWEN2.5-7B), and logical consistency was ver- ified by evaluating whether the model correctly derived the answer. The experimental results in Tab. 5 strongly sup- port the utility of this verification pipeline. Data routed through many pivot languages, including Tamil (93.6%), recorded reasoning accuracies com- parable to or even exceeding the baseline (90.0%). Even Kazakh, which showed relatively lower per- formance, maintained a decent accuracy of 89.2%, suggesting that information and logical structures essential for problem-solving were robustly pre- served without loss during the round-trip conver- sion to the target language. Of course, this round-trip reasoning accuracy does not perfectly guarantee the grammatical com- pleteness of the target language text itself. The pos- sibility that the high-performance reasoning model self-corrected minor noise in the back-translated text to derive the answer cannot be ruled out. How- ever, in low-resource language environments where securing native evaluators is realistically difficult, this is judged to be the most effective and scalable proxy metric for determining logical information loss. Detailed statistics of the finally constructed dataset are shown in Tab. 6. The dataset consists of a total of 8,500 high-quality samples, with final contributions by model being 62.1% for Gemini and 37.9% for GPT. StatisticValue Total Target Languages34 Samples per Language250 Total Samples8,500 Gemini Selection Ratio62.1% GPT Selection Ratio37.9% Avg. BT Similarity Score0.96 Table 6: Final statistics of the MGSM-LowResource dataset. D Linguistic Typology and Affinity/Alignment Classification To make the notions of surface affinity and struc- tural alignment used in Sec. 4.5 precise and repro- ducible, we define them operationally as functions of language-level typological features, listed for all target and skeleton languages in Tab. 7. Definitions. For a target languagetand a skele- ton language s: •Surface affinityAff(t,s)=1ifftandsshare at least one of: (i) genetic language family, (i) writing script, or (i) geographic macro- region; otherwise 0. • Structural alignmentAlign(t,s)=1ifftand sshare both (i) dominant morphological type (analytic, agglutinative, or fusional) and (i) basic constituent order (e.g., SOV vs. SVO); otherwise 0. These are descriptive categories, not predictors. As shown in Sec. 4.5, structural alignment does CodeLanguageFamilyMorph.Order Target languages kkKazakhTurkicAggSOV kyKyrgyzTurkicAggSOV mnMongolianMongolicAggSOV ugUyghurTurkicAggSOV hyArmenianIE (Armenian)Agg † SOV loLaoKra–DaiAnaSVO euBasqueIsolateAggSOV mtMalteseAfro-Asiatic (Semitic)FusSVO amAmharicAfro-Asiatic (Semitic)FusSOV neNepaliIE (Indo-Aryan)FusSOV siSinhalaIE (Indo-Aryan)FusSOV psPashtoIE (Iranian)FusSOV sdSindhiIE (Indo-Aryan)FusSOV paPunjabiIE (Indo-Aryan)FusSOV guGujaratiIE (Indo-Aryan)FusSOV mrMarathiIE (Indo-Aryan)FusSOV orOdiaIE (Indo-Aryan)FusSOV taTamilDravidianAggSOV mlMalayalamDravidianAggSOV Skeleton languages enEnglishIE (Germanic)Ana/FusSVO zhChineseSino-TibetanAnaSVO esSpanishIE (Romance)FusSVO ruRussianIE (Slavic)FusSVO koKoreanKoreanicAggSOV thThaiKra–DaiAnaSVO Table 7: Typological features of the 19 target and six skele- ton languages. Morph.: dominant morphological type (Ana: analytic/isolating; Agg: agglutinative; Fus: fusional). Order: basic constituent order. Scripts are listed in Tab. 4. † Armenian exhibits agglutinative nominal morphology. These features de- rive surface affinity and structural alignment as defined above. not guarantee positive transfer (e.g., Maltese is alignment-positive yet evaluation-sensitive), and surface affinity does not prevent negative transfer (e.g., ug–zh and mn–ru share affinity but are mis- aligned and transfer negatively). The classification removes ambiguity from our terminology rather than asserting a causal rule. Worked examples. Under these definitions, lo– th is both affine (Kra–Dai family) and aligned (an- alytic, SVO); eu–ko is not affine but is aligned (agglutinative, SOV); ug–zh and mn–ru are affine (geography or script) but not aligned. This matches the patterns reported in Sec. 4.5. E Computational Cost Analysis Fig. 4, 5 presents the quantitative analysis of the change in computational cost due to the introduc- tion of skeletons. The average length of the skeleton is highly concise, with an overall mean of 120.42 tokens. The average total token count (Skeleton + RS) is 1071.75, representing an increase of approx- imately 146.68 tokens compared to the vanilla base- line mean of 925.07. Notably, even in languages with high token consumption such as Swahili (sw), zheskothsw Language 110 112 114 116 118 120 122 124 Average Skeleton Token Count 118.4 120.8 121.7 121.4 119.9 Average Skeleton Token Count by Language Overall Mean: 120.42 Figure 4: Average token length of skeletons across query languages. zheskothsw 0 200 400 600 800 1000 1200 1400 1600 Token Count 815.5 953.8 905.7 991.6 1692.2 709.8 788.2 711.8 761.2 1654.4 Skeleton (Token + RS) vs Vanilla RS by Language Skeleton Total Mean: 1071.75 Vanilla Mean: 925.07 Skeleton (Token + RS) Vanilla RS Figure 5: Average generated token counts of the Qwen- series models. which requires 1692.2 tokens for the full process, the relative overhead remains manageable. These results suggest that the skeleton efficiently guides the model’s reasoning process, suppressing unnec- essary elaboration or inefficient operations during generation. F Qualitative Analysis F.1 Quality Assessment of English Skeletons Qwen2.5-7B Qwen2.5-14BQwen2.5-72BLlama-3.1-8B Llama-3.1-70B Model 0 2 4 6 8 10 Score Qwen FamilyLlama Family 5.85 5.61 6.22 5.24 6.06 5.91 4.83 5.88 4.63 6.03 7.45 7.50 7.90 7.05 7.97 6.40 5.98 6.67 5.64 6.69 (a) Skeleton Quality Scores by Model Overall (Avg)CompletenessSpecificityCoherence 0.00.20.40.60.81.0 Pearson Correlation (r) Completeness 0.74 Specificity 0.40 Coherence 0.82 Overall 0.63 (b) Correlation with PolyMath Accuracy Figure 6: Quality evaluation results for five models in the PolyMath Fig. 6 (Left) shows the change in skeleton gen- eration quality according to model scale. A dis- tinct trend of improved skeleton quality with larger ENZHESRUKOTH Skeleton Language MT TA GU EU MN LO KY UG Question Language 7.406.847.256.746.314.94 7.727.387.427.146.965.69 7.837.647.627.337.255.64 7.376.867.516.686.185.26 7.196.986.897.226.535.39 7.777.377.267.206.436.48 7.717.457.457.507.046.23 7.547.407.437.226.875.59 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 Score (Relative to Average) Figure 7: Skeleton quality heatmap across language pairs. English skeletons yield highest quality (7.57), while Thai performs worst (5.65). model scale was observed; in particular, large-scale models of 70B or larger recorded a high quality score of 6.7 on average. Furthermore, the correlation analysis results in Fig. 6 (Right) shed light on the relationship be- tween skeleton quality factors and final reasoning performance. ‘Coherence’ (r = 0.82) and ‘Com- pleteness’ (r = 0.74) showed strong positive cor- relations with final performance, whereas ‘Speci- ficity’ (r = 0.40) showed a relatively lower corre- lation. This implies that for successful reasoning in the target language, it is more important for the skeleton to include the logical flow of problem- solving and key elements without omission, rather than describing detailed steps specifically. F.2 Cross-Lingual Skeleton Quality and the Quality–Performance Gap The translation ablation in Sec. 5 controls skele- ton content to isolate surface form. Here we pro- vide complementary evidence from the opposite direction: we score the intrinsic generation quality of skeletons across languages using an LLM-as-a- Judge, and ask whether quality alone can explain the cross-lingual effects in Sec. 4.5. English dominates intrinsic quality.As shown in Fig. 7, English attains the highest average skeleton-quality score (7.57), followed by Spanish (7.35), Chinese (7.24), and Russian (7.13), while Korean (6.69) and Thai (5.65) score lower. This is consistent with English serving as a stable default skeleton language for most targets (Sec. 4.5). Quality does not track performance under struc- tural alignment.For several structurally aligned pairs, a lower-quality skeleton nonetheless yields higher performance. For Lao, the Thai skeleton scores1.29points below English in quality (6.48 vs.7.77) yet improves accuracy by+2.81; for Basque, the Korean skeleton scores1.19points below English (6.18vs.7.37) yet improves accu- racy by+6.32. A quality-only account would in- correctly predict English to be optimal here. No gap in the absence of structural alignment. Conversely, where the target and skeleton share surface affinity but not structural alignment, com- parable quality coincides with negative transfer. For Uyghur, the Chinese skeleton has quality compa- rable to English (7.40vs.7.54) but performance drops by−6.45; for Mongolian, the Russian skele- ton even slightly exceeds English in quality (7.22 vs.7.19) yet performance drops by−5.00. Follow- ing the affinity/alignment distinction in Sec. 4.5, these are precisely the pairs that share surface affin- ity (script or geography) but differ in morphological type and word order. Takeaway.Together with the translation ablation (Sec. 5), this indicates that skeleton-language ef- fects are not reducible to generation quality: struc- tural alignment can compensate for lower intrinsic quality, whereas its absence is not offset by high quality. We regard this as supporting evidence for treating skeleton language as a design variable to be identified through exploration, and—consistent with our framing throughout—as an exploratory hypothesis rather than a causal claim. F.3 Error Analysis: Impact of Skeleton LanguageMethodProblemConceptualReasoningCalculationOutput Gen. k COT-ℓ t 1006112933116 +SKELETON915410518135 ug COT-ℓ t 106741565795 +SKELETON1006013433135 mn COT-ℓ t 107781574788 +SKELETON1086214241115 hy COT-ℓ t 805710831155 +SKELETON88509522154 lo COT-ℓ t 1158215935179 +SKELETON1337515627123 eu COT-ℓ t 135881604776 +SKELETON140851553997 mt COT-ℓ t 112831544798 +SKELETON1246615448119 ne COT-ℓ t 77349323135 +SKELETON72448626139 Table 8: Comparison of COT-ℓ t and +SKELETON Results Across Different Reasoning Stages To investigate the underlying causes of the per- formance changes observed in Tab. 2, we con- ducted an in-depth error analysis on the entire eval- uation dataset using the LLM-as-a-Judge method- ology. Tab. 8 summarizes the frequency changes of the five key error types before and after applying the skeleton. The detailed evaluation protocol is described in Appendix B.2. To qualitatively analyze the effectiveness of the skeleton, we constructed an automated error clas- sification pipeline using thegpt-5-minimodel. For each incorrect sample, the model response was provided as input, and the prompt was designed to identify the most decisive error type among the following five categories: 1.Problem Comprehension Error: Misunder- standing the problem’s goal, constraints, or key variables. 2.Conceptual Error: Incorrect application or omission of necessary formulas, theorems, or scientific principles. 3.Reasoning Error: Errors in logical flow or inclusion of incorrect reasoning steps. 4.Calculation Error: Mistakes in arithmetic or symbolic operations despite a correct ap- proach. 5.Output Generation Error: Format errors (missing ), infinite repetition, lan- guage mismatch, etc. The evaluation model outputs the presence of each error type and a corresponding description in JSON format, which allowed for a quantitative comparative analysis of error patterns by language and condition. Skeletons as Cognitive ScaffoldingAnalysis of language groups with significant performance im- provement (k, ug, mn, hy) confirmed that the intro- duction of skeletons significantly suppresses Rea- soning Errors and Calculation Errors. In the case of Kazakh (k), which recorded the largest per- formance gain, Reasoning Errors decreased from 129 to 105 (−18.6%) and Calculation Errors plum- meted from 33 to 18 (−45.5%) after applying the skeleton. This suggests that the English skeleton serves as cognitive scaffolding that decomposes complex problems into manageable sub-units, ef- fectively preventing the model from deviating from the logical path or committing simple arithmetic mistakes during the reasoning process. Semantic Interference Causing Comprehension Failure Conversely, in language groups where Method MGSMMATH-500PolyMath zheskothswteAVG.zheskothswteAVG.zheskothswteAVG. Llama-3.1-8B-Instruct COT(ℓ q = target)71.574.758.665.561.756.264.733.237.729.831.031.925.131.523.624.926.219.327.616.223.0 + SKELETON68.375.561.065.163.857.465.234.735.634.032.034.225.932.724.126.324.421.129.116.823.6 COT(ℓ q = EN † )69.378.364.969.168.363.969.033.841.629.733.332.526.332.922.427.624.421.632.417.324.3 + SKELETON69.873.963.770.366.762.767.937.737.332.931.638.829.634.726.328.027.322.433.917.325.9 Llama-3.1-70B-Instruct COT(ℓ q = target)86.388.083.986.883.981.985.150.150.149.051.048.643.248.729.828.632.229.135.025.030.0 + SKELETON88.881.582.787.278.980.383.250.345.650.452.349.145.448.930.729.231.230.433.826.930.4 COT(ℓ q = EN † )85.588.484.785.183.383.185.049.550.951.947.251.146.249.529.432.932.928.736.326.031.0 + SKELETON85.982.383.982.782.582.783.351.150.951.249.048.245.649.329.229.731.728.336.327.330.4 Table 9: Multilingual mathematical reasoning results for the Llama 3.1 series (8B, 70B). The experiments use the MGSM, MATH-500, and PolyMath benchmarks.ℓ q = targetindicates that questions are presented in the target language, whileℓ q = EN † denotes the use of English-translated questions. The skeleton language is fixed to English in all cases (ℓ s = EN). ‘AVG.’ denotes the average over the six languages. performance improvement was marginal or de- clined (lo, eu, mt, ne), an increase in Problem Com- prehension Errors acted as the main failure factor. For Lao (lo), while the change in Reasoning Er- rors was minimal, Problem Comprehension Errors tended to increase from 115 to 133 (+15.7%). This is interpreted as a phenomenon where the English skeleton distorted the core requirements of the orig- inal problem due to semantic interference between languages, or the model became overly biased to- wards the English instructions of the skeleton, miss- ing the context of the original problem. G Llama 3.1 Experimental Results While the main text presented experimental results centering on the QWEN2.5 series, we performed the same experiments on the Meta LLAMA 3.1 series (8B, 70B) to verify the generalizability of the methodology. Tab. 9 summarizes the results of the MGSM, MATH-500, and PolyMath benchmarks for the LLAMA 3.1 model. The overall trends appeared similar to the QWEN2.5 series: •Improvement in Low-Resource Languages: In the CoT-en † setting of the LLAMA-3.1-8B model, a performance improvement from 32.5 to 38.8 (+6.3) was observed in MATH-500 for Swahili (sw) when skeletons were applied. •Diminishing Returns with Scale: In the LLAMA-3.1-70B model, the effect of skele- tons was minimal, or performance degradation occurred in some languages (es: 88.0→81.5). •Differences by Benchmark Difficulty: The most consistent improvement effects appeared in MATH-500, while MGSM and PolyMath showed large variations by language. Averaged over the six languages, Llama-3.1-8B shows a small positive skeleton effect, whereas the effect largely disappears or reverses for Llama- 3.1-70B (Tab. 9): the overall mean skeleton ef- fect is+0.81p (8B, target-language queries) and +0.75p (8B, English-translated queries), versus −0.43and−0.81p for 70B. The skeleton effect therefore varies across model variants and evalu- ation conditions rather than being uniformly pos- itive; what is shared with Qwen2.5 is the quali- tative tendency of the effect to shrink as model size and baseline performance increase (see also Appendix H). However, since Llama 3.1 has a rel- atively lower proportion of multilingual data, the baseline performance in low-resource languages was measured lower than that of Qwen2.5. Partic- ularly for Llama-3.1-70B, which already demon- strated high baseline performance in most con- ditions, the room for additional improvement by skeletons was limited. H Additional Model Family: Ministral ModelMGSMMATH-500PolyMathAVG.AVG. ∗ Ministral-3B+1.34+1.79+0.74+1.29+0.38 Ministral-14B−1.10+0.35+1.08+0.11−0.12 Table 10: English-skeleton effect∆(p) for the Minis- tral family under the CoT-ℓ t setting, averaged over the five languages of Tab. 1 (zh, es, ko, th, sw) per bench- mark. ‘AVG.’ denotes the average over all five lan- guages; AVG. ∗ excludes Swahili (sw), which retains relatively few valid samples. To examine whether the scale-dependent pattern is specific to Qwen2.5 and Llama 3.1, we addition- ally evaluate a third model family, MINISTRAL- INSTRUCT (3B and 14B) (Liu et al., 2026). To keep the comparison consistent, we fix the query condition to the target language (CoT-ℓ t ) and eval- uate both models on the three benchmarks and five languages of Tab. 1, yielding 30 cells.∆denotes the change in accuracy (p) from adding an English skeleton under the same query condition. As shown in Tab. 10, Ministral-3B improves by+1.29p on average, whereas the average ef- fect for Ministral-14B is only+0.11p; excluding Swahili (sw), which retains relatively few valid samples, the averages are+0.38and−0.12p, re- spectively. Across the 30 cells, baseline accuracy is negatively correlated with the skeleton effect (Spearmanρ = −0.568, nominalp = 0.0011). Thus, the English-skeleton effect is not consistently positive in this additional family either, and it varies across model variants and benchmarks. Because we evaluate only two sizes per family, we do not in- terpret this as a general scaling law; rather, across all three families we observe the same qualitative tendency that the average skeleton effect decreases as model size and baseline performance increase (mean∆under CoT-ℓ t : Qwen2.5+1.73→ +1.00 from 7B to 72B; Llama-3.1+0.81→−0.43from 8B to 70B; Ministral+1.29 → +0.11from 3B to 14B). These results distinguish the generality of the framework—the language decomposition (ℓ q ,ℓ s ,ℓ a )applies to any model—from the gen- erality of the observed performance effects, which do not automatically transfer across models. I PolyMath Difficulty-Level Analysis Model CoT-ℓ t CoT-en † Base +SKELETON Base +SKELETON QWEN2.5-7B4.754.91 (+0.16)5.125.82 (+0.70) QWEN2.5-14B6.628.40 (+1.78)7.137.56 (+0.43) QWEN2.5-72B9.279.97 (+0.70)8.928.96 (+0.04) Table 11: Skeleton gains on the two hardest PolyMath subsets (Top + High; 250 of 500 problems), as Exact-Match accuracy (%) averaged over five languages (zh, es, ko, th, sw). Parenthe- ses show the gain (∆) from adding the English skeleton. Com- pared with the full-set gains (Tab. 1), improvements shrink on the harder subset, consistent with the diminishing effect of skeletons as difficulty increases. PolyMath is organized into four difficulty levels (Low, Medium, High, Top; 125 problems each). To probe the effect of problem difficulty more directly than the full-set average, we restrict evaluation to the two hardest subsets (Top + High; 250 problems) and recompute the English-skeleton gain for the QWEN2.5 family (7B, 14B, 72B), averaged over five languages (zh, es, ko, th, sw). Tab. 11 reports the results. Compared with the gains aggregated over all 500 problems, improvements on the harder sub- set generally shrink: for QWEN2.5-7B the average CoT-ℓ t gain drops from+1.20to+0.16, and for QWEN2.5-72B from+0.90to+0.70. The mid- sized QWEN2.5-14B is an exception, showing a larger gain (+1.78) on the harder subset, suggest- ing that skeleton guidance can still aid mid-sized models on difficult problems. Overall, the tendency is consistent with our main finding that the ben- efit of skeletons diminishes as problem difficulty increases. J Statistical Significance Analysis TargetSkeleton∆ (%)p Positive effects (Non-English skeleton > English skeleton) mtru+8.620.001 mtth+6.900.008 mtes+6.610.041 Negative effects (English skeleton > Non-English skeleton) ugth−10.250.002 mnth−8.470.016 kkko−7.660.030 Table 12: Pairs that reached nominal significance in Mc- Nemar’s test before correction for multiple comparisons (p < 0.05).∆denotes the accuracy difference between the non-English skeleton and the English skeleton. To assess the statistical reliability of the non- English skeleton effects discussed in Sec. 4.5, we conducted McNemar’s tests. McNemar’s test com- pares paired binary outcomes over the same set of instances, focusing on the transition patterns be- tween correct and incorrect predictions under two conditions. In our analysis, we tested whether item- level correctness differed significantly between the English skeleton and each non-English skeleton. Using the greedy decoding results of QWEN2.5- 7B, we examined a total of 95 target–skeleton pairs, corresponding to 19 low-resource target languages and five non-English skeleton languages. Results. Among the 95 pairs, six reached nomi- nal significance before correction for multiple com- parisons (p < 0.05). These include both positive effects, where a non-English skeleton outperformed the English skeleton, and negative effects, where the English skeleton outperformed the non-English skeleton. The three positive effects were all observed for Maltese (mt), with mt–ru showing the strongest sig- nal (p = 0.001). In contrast, negative effects were observed for Uyghur–Thai, Mongolian–Thai, and Kazakh–Korean. These results indicate that non- English skeletons can also significantly degrade performance relative to English skeletons in some cases. The pairs discussed in the main text as direc- tionally consistent cross-lingual effects, namely eu– ko (p = 0.072), hy–ko (p = 0.361), and lo–th (p = 0.401), do not reach nominal significance un- der McNemar’s test. We therefore do not interpret them as effects confirmed by a single statistical test. Instead, we treat them as consistency-based observations, since the direction of the effect re- mains stable across greedy decoding and multi- rollout evaluation. In other words, McNemar’s test examines item-level correctness transitions under a single decoding setting, whereas our robustness analysis evaluates whether the direction of the ef- fect is preserved across evaluation protocols. For Maltese, the positive effects are nominally significant under greedy decoding, but they are sub- stantially attenuated or even reversed under multi- rollout evaluation. This is consistent with our inter- pretation in the main text that Maltese represents an evaluation-sensitive case rather than a directionally consistent transfer pattern; the cross-benchmark re- sults in Appendix K further show that the Maltese gains are benchmark-dependent. English Skeleton vs. No Skeleton. Following the same procedure, we also tested whether En- glish skeletons significantly outperform the no- skeleton CoT baseline on the 19 low-resource MGSM languages (QWEN2.5-7B, CoT-ℓ t ). The effect was positive for 13 of the 19 languages, with a mean language-level change of+2.38p. Af- ter FDR correction, however, significant improve- ments remained for only two languages: Kazakh (k;40.08 → 51.42,∆ = +11.34,p FDR = 0.014 ) and Uyghur (ug;27.13 → 37.65,∆ = +10.52,p FDR = 0.014). The−5.65p decrease for Maltese was nominally significant before cor- rection (p = 0.049) but not after FDR correction (p FDR = 0.307), and no other language showed a significant difference after correction. When the 19 languages are pooled at the item level (n = 4,628), the overall effect is significant (∆ = +2.27p, p = 0.0014). In other words, the benefit of En- glish skeletons is real but too small to detect at the individual-language level, and it is driven by a few languages such as k and ug. K Cross-Benchmark Generalization on MSVAMP TargetSkel.MGSM ∆MSVAMP ∆Outcome hyko+5.17+2.75Retained lozh+3.63+2.63Retained loth+2.81+2.42Retained euko+6.320.00Disappeared mtes+6.61+2.00Attenuated mtth+6.90+0.43Attenuated mtru+8.62−1.27Reversed ugth−10.25−2.14Retained (neg.) mnth−8.47−2.53Retained (neg.) kkko−7.66−4.63Retained (neg.) Table 13: Cross-benchmark transfer of skeleton- language effects from MGSM to MSVAMP (QWEN2.5- 7B, CoT-ℓ t ).∆: accuracy difference (p) between the non-English and the English skeleton. Pairs were fixed using only the MGSM results before inspecting MSVAMP. Across all 35 target–skeleton cells, the sign is preserved in 23/35 (66%). To test whether skeleton-language effects ob- served on MGSM generalize to an unseen test set, we evaluate on MSVAMP (Chen et al., 2024), a separate benchmark whose problems do not over- lap with MGSM (up to 1,000 mathematical rea- soning problems per language). Since MSVAMP natively supports only ten languages, we obtained additional low-resource target languages using the same translation and verification pipeline described in Appendix C, extending it to eight languages (ko, eu, hy, lo, mn, ug, mt, and k). We use the seven target languages for which MGSM-LowResource results are available (eu, hy, lo, mn, ug, mt, and k). Protocol.To rule out post-hoc selection, the eval- uation targets, selection rule, and skeleton candi- dates were fixed using only the MGSM results be- fore inspecting MSVAMP. We selected eu, hy, and lo (directionally consistent pairs in Sec. 4.5), mn and ug (asymmetric-negative cases), and mt and k (nominally significant results in Appendix J). For each target language, we applied the five non- English skeleton languages (zh, es, ru, ko, and th) under the target-language query condition and com- puted∆, the accuracy difference relative to the English skeleton, on both benchmarks, yielding 7× 5 = 35 target–skeleton cells. Results. Across the 35 cells, effect sizes on MGSM and MSVAMP show a moderate positive rank correlation (Spearmanρ = 0.383, nominal p = 0.023), and the sign of the effect is preserved in 23 of the 35 cells (66%; binomialp = 0.090). Tab. 13 summarizes the key pairs. Two observations stand out. First, the asymmetric-negative pattern transfers relatively well: among the 15 cells for mn, ug, and k, 14 remain negative on MSVAMP, with only ug–ru reversing (−4.11→ +1.79). Sec- ond, the positive Maltese effects are not robust: mt– es remains positive (+6.61 → +2.00), but mt–th largely diminishes (+6.90→ +0.43) and mt–ru re- verses (+8.62→−1.27). The transfer of positive selections is mixed: hy–ko, lo–zh, and lo–th retain their positive direction, whereas eu–ko vanishes to 0.00. Implications.We therefore do not claim that the optimal skeleton language can be specified in ad- vance for an arbitrary target language. Within the evaluated subset, results from an existing bench- mark are more useful for identifying potentially risky skeleton candidates and prioritizing subse- quent evaluations than for directly determining an optimal skeleton language. This asymmetry is an exploratory observation, and separate validation re- mains necessary for a new benchmark, model, or task. As MGSM and MSVAMP focus on elemen- tary arithmetic, testing transfer on harder bench- marks is left for future work. L Model Performance Across Other Low-Resource Languages L.1 Results on Telugu Tab. 14 presents the performance changes when applying the skeleton to Telugu. L.2 Results on All Low-Resource Languages Although our experiments cover 34 languages in total, target-language CoT (COT-ℓ t ) fails to oper- ate reliably for a subset of languages, making fair comparison difficult. We therefore focus our main analysis on 19 languages where target-language reasoning is stable. For completeness and trans- parency, results for all 34 languages are reported in Fig. 8. MethodMATH-500 (te)PolyMath (te)MGSM (te) Qwen2.5-7B-Instruct CoT-ℓ t 29.812.230.40 +SKELETON27.812.433.60 CoT-en † 35.717.360.40 +SKELETON33.917.754.00 Qwen2.5-14B-Instruct CoT-ℓ t 43.418.654.80 +SKELETON45.919.855.20 CoT-en † 45.124.271.89 +SKELETON43.022.273.90 Qwen2.5-72B-Instruct CoT-ℓ t 53.223.765.20 +SKELETON55.427.468.00 CoT-en † 52.428.278.40 +SKELETON55.228.882.80 Llama-3.1-8B-Instruct CoT-ℓ t 25.816.256.40 +SKELETON25.016.857.60 CoT-en † 26.717.363.60 +SKELETON30.117.362.80 Llama-3.1-70B-Instruct CoT-ℓ t 43.625.082.00 +SKELETON45.926.980.40 CoT-en † 46.726.083.20 +SKELETON46.127.382.80 Table 14: Telugu (te) Accuracy (%) on MATH-500, PolyMath, and MGSM ZH (Chinese)ES (Spanish)RU (Russian)KO (Korean)TH (Thai) Skeleton Language TA KN MY KM AM YO SI GU NE UZ KY CEB EU GN HY JV KA K KU LO MG ML MN MR MT OR PA PS QU SD SO SU TG UG CoT Language (question_language) N=250 +0.0% N=250 -5.2% N=250 -2.4% N=212 -6.6% N=249 -4.4% N=249 -1.2% N=250 -18.8% N=250 +0.4% N=184 -2.2% N=246 +6.1% N=250 +0.8% N=204 -2.9% N=250 -2.8% N=50 +10.0% N=247 -0.8% N=191 +4.7% N=194 +1.0% N=181 +6.6% N=51 +7.8% N=157 +3.2% N=240 -2.1% N=228 -0.4% N=241 -2.1% N=234 +1.7% N=235 -1.7% N=89 +2.2% N=91 +5.5% N=91 +3.3% N=10 +0.0% N=94 +3.2% N=247 -0.4% N=246 -0.8% N=246 -0.4% N=226 -2.2% N=244 -1.6% N=250 -0.8% N=250 -4.4% N=247 -5.3% N=238 -5.0% N=250 -4.0% N=240 +2.5% N=244 -1.2% N=242 -0.8% N=223 -0.9% N=230 -0.9% N=80 +7.5% N=109 +0.0% N=24 -8.3% N=31 -19.4% N=54 +0.0% N=240 -2.5% N=240 +1.7% N=209 +0.0% N=201 -7.0% N=248 -0.8% N=130 -0.8% N=133 -1.5% N=144 -2.1% N=76 +1.3% N=145 -5.5% N=232 +1.7% N=197 -2.0% N=235 +0.0% N=174 +6.3% N=207 +0.5% N=55 +5.5% N=37 -5.4% N=45 +0.0% N=50 +4.0% N=62 +1.6% N=249 -4.4% N=246 +1.2% N=246 -2.0% N=116 +4.3% N=246 -6.5% N=65 +7.7% N=53 +5.7% N=15 +6.7% N=60 +5.0% N=249 -4.4% N=226 -2.7% N=247 -0.4% N=38 +0.0% N=248 -4.4% N=242 -3.3% N=247 -1.6% N=210 +0.5% N=209 -7.7% N=247 -3.6% N=248 +3.6% N=250 -2.8% N=250 +1.6% N=243 +2.5% N=249 +2.8% N=233 -1.7% N=220 +2.3% N=218 +0.9% N=238 +3.4% N=249 +2.8% N=249 -2.4% N=245 -1.6% N=156 -1.3% N=249 -4.4% N=241 -4.6% N=217 -5.5% N=200 -5.0% N=228 -0.9% N=236 -8.5% N=247 +3.6% N=247 +1.6% N=244 +2.0% N=148 +0.7% N=246 -0.4% N=240 +3.8% N=227 +6.6% N=232 +9.1% N=213 +1.9% N=232 +7.3% N=249 +0.0% N=247 -1.6% N=242 +3.7% N=246 -1.6% N=247 -3.2% N=249 -1.2% N=249 -3.2% N=249 -2.8% N=192 +0.0% N=249 -4.0% N=231 -1.7% N=226 -2.7% N=229 +0.0% N=165 -0.6% N=229 -0.9% N=135 +1.5% N=107 -0.9% N=149 -3.4% N=90 -2.2% N=146 -5.5% N=213 -3.3% N=190 -4.2% N=193 -1.0% N=171 -1.2% N=185 +0.5% N=37 +2.7% N=37 +8.1% N=45 +0.0% N=35 +5.7% N=50 +2.0% N=99 -3.0% N=68 -14.7% N=83 +0.0% N=39 -5.1% N=89 +2.2% N=168 -4.8% N=164 -0.6% N=67 -6.0% N=142 -2.1% N=180 -8.3% N=248 -6.5% N=243 -4.5% N=219 -4.1% N=198 -6.1% N=244 -10.2% Skeleton Language Performance Delta Heatmap Qwen2.5-7B-Instruct (Non-English Skeleton - English Skeleton) 15 10 5 0 5 10 15 Accuracy Delta (%) Figure 8: Performance differences (Non-English English) under the CoT-ℓ t setting using QWEN2.5-7B across all 34 target languages. While English generally serves as a stable default skeleton language, several target languages exhibit gains with specific Non-English skeletons, indicating that skeleton effectiveness is language-dependent.