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BoundaryEvo: Data-Model Co-Evolution at the Capability Boundary for Long-Chain Reasoning Distillation

Jiaming Su, Suorong Yang, Xudong Li, Tengchao Yang, Peng Ye

Year: 2026Venue: arXiv preprintArea: cs.AIType: PreprintEmbeddings: 111

Intelligence

Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 91%

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Summary

The paper introduces BoundaryEvo, a two-stage data-model co-evolution framework for distilling long-chain reasoning from large teacher models to small student models. Unlike static distillation, BoundaryEvo adapts training data to the student's evolving capability boundary. Stage 1 (Capability Priming) establishes initial reasoning skills via judge-free direct distillation. Stage 2 (Capability-Boundary Co-Evolution) iteratively selects questions that the expert can solve but the student cannot, ensuring high-quality supervision. Evaluated on Qwen3.5-4B with only 1,000 training examples, it significantly outperforms static and dynamic baselines across 11 reasoning benchmarks.

Entities (10)

BoundaryEvo · method · 98%Qwen3.5-4b · model · 95%AMO-Bench · benchmark · 92%IMO-AnswerBench · benchmark · 92%AIME'25 · benchmark · 90%Capability-Boundary Co-Evolution · process · 90%Capability Priming · process · 90%HMMT25 · benchmark · 90%DeepSeek-V3.2-Speciale · model · 85%GPT-5.1 · model · 85%

Relation Signals (6)

BoundaryEvo → improves → Qwen3.5-4b

confidence 95% · raising the original Qwen3.5-4B accuracy from 62.9 to 75.3 on IMO-AnswerBench

BoundaryEvo → uses → Capability Priming

confidence 95% · In Capability Priming, we directly distill a small set of teacher-generated long reasoning trajectories

BoundaryEvo → uses → Capability-Boundary Co-Evolution

confidence 95% · In Capability-Boundary Co-Evolution, we repeatedly select questions that the current student has not mastered

BoundaryEvo → outperforms → static and dynamic data selection baselines

confidence 90% · BoundaryEvo consistently outperforms static and dynamic data selection baselines across 11 reasoning benchmarks

DeepSeek-V3.2-Speciale → generates → Reasoning Trajectories

confidence 85% · reasoning trajectories re-distilled from DeepSeek-V3.2-Speciale

GPT-5.1 → provides → expert rollouts

confidence 85% · GPT-5.1 (OpenAI, 2025) provides expert rollouts for capability-boundary estimation

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Abstract

Abstract:Distilling long-chain reasoning trajectories from large teacher models provides an effective way to improve the reasoning ability of small language models. However, existing methods typically select a fixed training set before distillation, overlooking that the usefulness of a reasoning example changes as the student evolves. This mismatch becomes especially costly when only a small number of teacher trajectories can be used for training. Besides, examples already mastered provide little new supervision, while examples far beyond the capability of student may be difficult to learn from. Training data should therefore adapt as model capability evolves, yet reliable data selection is difficult when the student is initially weak. To address these challenges, our key idea is to first establish the basic reasoning capability and then adapt the training data to the evolving capability boundary of the student. We propose BoundaryEvo, a two-stage data--model co-evolution framework. In Capability Priming, we directly distill a small set of teacher-generated long reasoning trajectories without using a separate model to judge and filter them, allowing the student to acquire an initial reasoning capability before capability-aware data selection. In Capability-Boundary Co-Evolution, we repeatedly select questions that the current student has not mastered but the expert can reliably solve, and update the training data as the student evolves. With only 1,000 training examples, BoundaryEvo consistently outperforms static and dynamic data selection baselines across 11 reasoning benchmarks, raising the original Qwen3.5-4B accuracy from 62.9 to 75.3 on IMO-AnswerBench and from 26.8 to 46.0 on AMO-Bench. The code and data will be made publicly available.

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Preprint. Under Review. BOUNDARYEVO: DATA-MODEL CO-EVOLUTION AT THE CAPABILITY BOUNDARYFOR LONG-CHAIN REASONING DISTILLATION Jiaming Su 1,∗ , Suorong Yang 2,∗ , Xudong Li 1,∗ , Tengchao Yang 3 , Peng Ye 4,† 1 Shanghai Artificial Intelligence Laboratory 2 Nanjing University 3 Tongji University 4 The Chinese University of Hong Kong * Equal contribution † Corresponding authors ABSTRACT Distilling long-chain reasoning trajectories from large teacher models provides an effective way to improve the reasoning ability of small language models. However, existing methods typically select a fixed training set before distillation, overlooking that the usefulness of a reasoning example changes as the student evolves. This mismatch becomes especially costly when only a small number of teacher trajecto- ries can be used for training. Besides, examples already mastered provide little new supervision, while examples far beyond the capability of student may be difficult to learn from. Training data should therefore adapt as model capability evolves, yet reliable data selection is difficult when the student is initially weak. To address these challenges, our key idea is to first establish the basic reasoning capability and then adapt the training data to the evolving capability boundary of the student. We propose BoundaryEvo, a two-stage data–model co-evolution framework. In Capability Priming, we directly distill a small set of teacher-generated long rea- soning trajectories without using a separate model to judge and filter them, allowing the student to acquire an initial reasoning capability before capability-aware data selection. In Capability-Boundary Co-Evolution, we repeatedly select questions that the current student has not mastered but the expert can reliably solve, and update the training data as the student evolves. With only 1,000 training examples, BoundaryEvo consistently outperforms static and dynamic data selection baselines across 11 reasoning benchmarks, raising the original Qwen3.5-4B accuracy from 62.9 to 75.3 on IMO-AnswerBench and from 26.8 to 46.0 on AMO-Bench. The code and data will be made publicly available. Data Evolution (a) Static Distillation Verified Trajectory Teacher Seed QA Low Performance Stage1: Capability Priming Stage2: Capability-Boundary Co-Evolution Boundary Awareness Boundary Awareness Data Evolution (c) BoundaryEvo (Ours) Seed QA SFT (b) Error-Driven Evolution Expanded Data Similar Questions Student Errors Teacher Data Evolution Boundary Awareness (e.g. LLM2LLM) Student t M Teacher Selected Boundary Data Student SFT Evolution QA Pool Multi Agent System Boundary Trajectory SFT Raw Trajectory Teacher Priming QA SFT Primed Student 0 M (d) Performance (%) 76 80 84 88 92 AIME25 92.5 79.8 84.0 90.7 83.3 81 84 87 90 93 AIME26 93.3 84.0 86.7 90.0 88.3 70 75 80 85 90 95 HMMT25 90.0 73.8 85.6 84.3 80.0 68 72 76 80 84 HMMT26 82.3 68.9 74.8 72.772.7 75 80 85 90 95 100 BruMO25 95.8 83.5 79.7 86.7 85.0 60 64 68 72 76 IMO-Ans 75.3 62.9 66.2 67.4 61.6 24 30 36 42 48 AMO-Bench 46.0 26.8 39.0 39.8 26.5 8 16 24 32 40 ArxivMath 35.9 25.2 25.9 28.2 9.7 66 72 78 84 90 96 ZebraLogic 96.0 92.4 93.9 95.6 69.0 66 69 72 75 78 81 GPQA-D 81.3 76.2 78.1 79.3 68.4 40 44 48 52 56 SuperGPQA 55.0 52.9 52.8 53.8 42.9 BoundaryEvo-4BQwen3.5-4BQwen3.5-4B + RandomQwen3.5-4B + LLM2LLMDASD-4B-Thinking Figure 1: BoundaryEvo evolves supervision along the capability boundary of the student model. (a-c) Comparison with static distillation and error-driven acquisition, BoundaryEvo dynam- ically adapts supervision according to the evolving capability boundary of the student model. (d) BoundaryEvo-4B consistently outperforms strong baselines across various reasoning benchmarks. 1 arXiv:2610.13285v1 [cs.AI] 9 Oct 2026 Preprint. Under Review. 1INTRODUCTION Distilling long-chain reasoning trajectories from large teacher models is an effective way to transfer complex reasoning capabilities to small student models (DeepSeek-AI, 2025; Moshkov et al., 2025). Yet reasoning supervision is typically constructed once (Han et al., 2025; Moshkov et al., 2025) and then treated as static throughout training, even though the learner itself is continuously changing. As shown in Figure 1(a), existing distillation pipelines typically generate and filter a fixed collection of teacher trajectories before optimization and repeatedly train on the same data thereafter (Yan et al., 2026). Such a static mechanism overlooks a fundamental property of learning: data utility is non-stationary with respect to the evolving student. Examples that once exposed meaningful capability gaps may later become redundant, whereas trajectories that were initially beyond the student’s reach may become precisely the supervision needed for its next stage of improvement. Thus, the supervision itself should continually adapt to the student’s evolving capability boundary, enabling data and model to co-evolve throughout learning. Recent efforts have begun to adapt training data to the evolving student, as illustrated in Figure 1(b), typically by prioritizing questions that the current model fails to solve (Lee et al., 2024). While this moves beyond static supervision, such adaptation remains largely reactive: it responds to the student’s current errors without explicitly modeling how its capability changes over the course of learning. As a result, the training data may shift with the student, but it does not necessarily co-evolve with the student’s emerging learning needs. For a weak student, failures may span questions that are only slightly beyond its current capability as well as those that remain far out of reach, while student difficulty alone cannot guarantee useful supervision either Tzannetos et al. (2023); Song et al. (2020); Chen et al. (2025). Thus, data-model co-evolution requires more than tracking what the student gets wrong. It requires continually localizing the capability boundary where unresolved student needs align with reliable and informative teacher supervision. Based on this insight, we introduce BoundaryEvo, a two-stage data–model co-evolution framework for long-chain reasoning distillation under a limited supervision budget. As shown in Figure 1(c), BoundaryEvo organizes reasoning distillation as a two-stage co-evolution process: it first establishes a sufficiently capable student state and subsequently tracks the evolving capability boundary between the student and fixed expert to guide supervision. In Stage 1, Capability Priming, the student is initial- ized through direct distillation of teacher-generated reasoning trajectories. Rather than specializing supervision to a prematurely estimated learning boundary, this stage exposes the student to a broad dis- tribution of teacher reasoning behaviors, providing the capability foundation required for subsequent boundary-aware adaptation. In Stage 2: Capability-Boundary Co-Evolution, we repeatedly identify questions that remain unresolved by the current student but can be reliably solved by the fixed expert, and then acquire teacher trajectories for the selected questions. Following each student update, the capability boundary is re-estimated and the supervision set is correspondingly refreshed, forming a closed-loop interaction between model learning and data acquisition. This iterative coupling progres- sively redirects supervision toward the student’s current learning frontier with reduced supervision budget, allowing the training data and model capability to co-evolve throughout distillation. With a total budget of only 1,000 training examples, BoundaryEvo consistently outperforms strong static and dynamic data selection baselines on Qwen3.5-4B across 11 reasoning benchmarks (Figure 1(d)). Compared with the original Qwen3.5-4B model, BoundaryEvo improves accuracy from 62.9 to 75.3 on IMO-AnswerBench and from 26.8 to 46.0 on AMO-Bench. Experiments on Gemma-4-E4B-it further show that the improvements transfer across model backbones. Our contributions are threefold: • We identify a stage-dependent principle for long-chain reasoning distillation. Early-stage capability priming benefits from retaining diverse teacher reasoning behaviors, including exploration and unsuccessful attempts, rather than filtering trajectories solely by final correctness, while later-stage training benefits from capability-boundary-guided data selection. •We propose BoundaryEvo, a two-stage data–model co-evolution framework for long-chain rea- soning distillation, which first establishes student capability through capability priming and then adaptively evolves long-horizon supervision through student–expert capability-boundary tracking. •We evaluate BoundaryEvo with only 1,000 training examples across 11 reasoning benchmarks and multiple model backbones, showing consistent gains over static and dynamic data-selection baselines and validating the roles of both capability priming and capability-boundary co-evolution. 2 Preprint. Under Review. 2RELATED WORK Reasoning distillation. Knowledge distillation transfers teacher predictions or generated sequences to student models (Hinton et al., 2015; Kim & Rush, 2016). Chain-of-thought and self-consistency demonstrate the value of intermediate reasoning trajectories (Wei et al., 2022; Wang et al., 2023a). Recent reasoning models, including DeepSeek-R1 (DeepSeek-AI, 2025), O1 replication (Huang et al., 2024), and DASD (Yan et al., 2026), further highlight the effectiveness of long-chain reasoning supervision. LIMO (Ye et al., 2025) and s1 (Muennighoff et al., 2025) achieve strong performance with small curated datasets, while OpenMathReasoning studies large-scale mathematical reasoning data construction (Moshkov et al., 2025). VibeThinker (Xu et al., 2026) explores multi-stage reasoning training beyond single-step distillation. However, they largely assume a static supervision distribution and overlook the need to adapt training data as model capabilities evolve. Self-evolution and co-evolution. Self-improvement methods iteratively generate experience, obtain feedback, and update models. STaR (Zelikman et al., 2022) and ReST (G ̈ ulc ̧ehre et al., 2023) alternate reasoning generation and training with feedback signals. Self-rewarding models improve feedback generation through model-based rewards (Yuan et al., 2024). R-Zero (Huang et al., 2025) , Socratic-Zero (Wang et al., 2025), and SeRL (Fang et al., 2025) explore self-play and challenger– solver evolution strategies. SyncLoop (Chen et al., 2026) studies multimodal data evolution coupled with model improvement. Synthetic data approaches, including Self-Instruct (Wang et al., 2023b), WizardLM (Xu et al., 2024), WizardMath (Luo et al., 2025), and MetaMath (Yu et al., 2024), construct additional instruction or reasoning data. AutoData (Kulikov et al., 2026) introduces an agentic pipeline for task generation and refinement, while LLM2LLM (Lee et al., 2024) selects teacher-generated data based on student errors. Different from these methods, BoundaryEvo performs capability-boundary-driven data–model co-evolution. Data selection. Training data selection has been explored through curriculum learning, active learning, and quality-based filtering. Curriculum learning and active learning connect sample selection with learner states (Bengio et al., 2009; Ren et al., 2022; Yang et al., 2026). LIMA (Zhou et al., 2023), DEITA (Liu et al., 2024), QuRating (Wettig et al., 2024), and educational-data studies (Gunasekar et al., 2023) investigate data quality, diversity, and selection strategies. D4 (Tirumala et al., 2023) studies deduplication, diversity, and repeated data utilization. Experience replay provides a general mechanism for retaining previous knowledge during iterative learning (Mnih et al., 2015; Schaul et al., 2016). Outcome and process verification introduce additional signals for reasoning data assessment (Cobbe et al., 2021; Lightman et al., 2024; Uesato et al., 2022; Wang et al., 2024). Although model- based judging is useful, it can still be imperfect (Zheng et al., 2023). BoundaryEvo selects data based on the capability boundary between student and expert, rather than static quality or error signals. 3THE PROPOSED BOUNDARYEVO Overview. We propose BoundaryEvo, a two-stage data–model co-evolution framework that dynami- cally adapts training data to the evolving capabilities of compact reasoning models. As illustrated in Figure 2, Stage 1 performs direct distillation from a small set of teacher-generated reasoning trajec- tories without additional model-based judging, providing the student with foundational reasoning capabilities. Stage 2 then iteratively identifies capability-boundary examples: at each evolution turnt, the current student and a fixed expert modelEindependently solve the same candidate questions and generate reasoning trajectories, respectively. An analyzer evaluates student performance, expert solutions, and data quality to select questions that the expert can reliably solve but the student has not yet mastered. The selected questions are combined with revalidated historical data, and verified teacher trajectories are distilled through supervised fine-tuning to update the student model. As the student improves, the capability boundary shifts and guides subsequent data selection, enabling continuous data–model co-evolution under a fixed data budget. 3.1STAGE 1: JUDGE-FREE DIRECT DISTILLATION FOR CAPABILITY PRIMING Given seed questionsx i K 0 i=1 , the teacher model generates reasoning trajectories and final answers: y T i ∼ T (·| x i ), D 0 =(x i ,y T i ) K 0 i=1 .(1) Stage 1 performs direct reasoning distillation, where teacher-generated trajectories are used as supervision after basic response-validity checks. Unlike the subsequent evolution stage, Stage 1 does 3 Preprint. Under Review. Expert Response Boundary QA next turn Capability Analyzer •Clear •Answerable •Context Support Validity Checks Capability Gap studentert S exp 60.030.090.0 Scoring & Analysis Question & Answer Student Response Expert Response Student 1t M 0 M Student t M Trajectory Trajectory ... Evolution QA Pool (2k) Expert Student Response SFT Teacher Rollout w/ verification Teacher Teacher Trajectory Retain the trajectory of correct answers Boundary Trajectory Priming QA Set (0.1k) Teacher Rollout w/o verification Teacher Answer the Question step by step ... question... Trajectory Capability Priming Training Student Model Initial model SFT Answer the Question step by step ... question... Answer the Question step by step ... question... Student Correctness 0.30 Expert Correctness 0.90 Data Quality 0.85 Answer the Question step by step ... question... Selected Boundary QA Student Can Solve Capability Boundary Too Difficult Expert Can Solve Stage 1: Capability Priming Stage 2: Capability-Boundary Co-Evolution Figure 2: Overview of BoundaryEvo. Stage 1 performs judge-free teacher distillation to obtain the capability-primed checkpoint. In Stage 2, responses from the current student and a fixed expert are used to judge the candidate QAs, from which replay and novel records are selected for multiple teacher rollouts. The resulting trajectories are then independently verified against reference answers, and only those that pass verification supervise the next student. not apply additional model-based judging or capability-gap-based filtering. Instead of searching for difficult examples at this stage, it aims to efficiently transfer teacher reasoning behaviors and establish initial student reasoning capabilities from a small amount of high-quality supervision. This design avoids premature filtering that may discard useful teacher experiences due to imperfect automated assessment or diverse valid reasoning patterns. Training onD 0 produces the capability- primed checkpointM 0 , which provides a stronger initialization for Stage 2. With this initial capability, BoundaryEvo can more effectively identify capability-boundary questions and perform subsequent verified teacher trajectory distillation for iterative data–model co-evolution. 3.2STAGE 2: DYNAMIC CAPABILITY-BOUNDARY CO-EVOLUTION Student–Expert–Analyzer rollouts and judging. At evolution roundt, the student modelM t and a fixed expert independently generate reasoning trajectories for the same candidate question x i : ˆy s i,t ∼ M t (·| x i ,h i ),ˆy e i ∼ E(·| x i ,h i ).(2) Here,h i denotes optional context or auxiliary information. The analyzer evaluates the question, reference information, and reasoning outputs from both models. The resulting assessment captures three aspects: capability-boundary matching, data-quality control, and question-validity checks: a (t) i =    s s,(t) i ,s e i ,g (t) i |z Boundary matching , q i |z Quality ,b clear i ,b answerable i ,c i | z Validity    .(3) The capability gap is computed from the student and expert scores: g (t) i = s e i − s s,(t) i .(4) A large positive gap indicates that the expert can solve the problem beyond the current student capability. However, capability difference alone is insufficient, as the expert response must be reliable and the example must provide high-quality and valid supervision. The quality scoreq i measures 4 Preprint. Under Review. overall training suitability, while validity indicatorsb clear i ,b answerable i , andc i prevent ambiguous questions or unsupported contexts from being mistaken as student limitations. Hard capability-boundary filtering. The three levels define the acceptance criteria for capability- boundary examples. A candidate is selected intoA t only if it satisfies all three conditions: Boundary matching: s e i ≥ τ e , s s,(t) i < τ s , g (t) i ≥ τ g , Quality control: q i ≥ τ q , Question validity: b clear i = b answerable i = 1, c i ≥ τ c . (5) The first condition identifies examples where the expert provides reliable solutions while the student can improve. The remaining conditions ensure that selected samples provide high-quality and valid supervision rather than reflecting ambiguity or data defects. Together, these criteria identify training examples near the current capability boundary. BoundaryEvo therefore selects not only examples that the student fails but also those that can provide reliable learning signals for evolution. Replay and novel acquisition. To balance retaining unresolved examples and exploring new boundary samples, we maintain a fixed training budget ofK = 300examples per turn. At each round, the updated student rescans the evolution poolPof 2K candidates, producing the accepted set A t . We divide it into replay examples previously used inD t−1 and novel examples: R t =z ∈A t : x(z)∈ Questions(D t−1 ), N t =z ∈A t : x(z) /∈ Questions(D t−1 ). (6) Our replay-first strategy keeps eligible replay examples up to the replay quota and fills the remaining budget with quality-ranked novel examples: D t =R ′ t ∪ Select K−|R ′ t | (N t ), |D t | = K = 300.(7) Novel examples are ranked by student response completeness, analyzer recommendation, and quality scorer (t) i , with secondary criteria used for ties. This design preserves unresolved learning signals while continually introducing new capability-boundary examples. Post-selection teacher rollouts and verified distillation. After QA selection, we construct supervi- sion through fresh teacher rollouts. This separates which questions to learn from which solutions to imitate: the expert rollout identifies the capability boundary, while the teacher rollouts provide high-quality reasoning trajectories for student training. For each selected QA recordz i ∈ D t , the teacher independently samples L≥ 2 reasoning trajectories: y T,(k) i,t ∼ T (·| x i ,h i ), k = 1,...,L.(8) We retain trajectories whose final answers pass reference verification: V (Ans(y T,(k) i,t ),y ref i ) = 1.(9) A verified trajectory is selected for each question to formD distill t , and the student is fine-tuned on these full reasoning trajectories. Unlike Stage 1 judge-free priming, Stage 2 requires evaluation before distillation because data selection now depends on identifying capability-boundary examples rather than directly transferring teacher behaviors. QA filtering identifies learning opportunities, while verification ensures reliable training targets for the next evolution round. 3.3TRAINING OBJECTIVE Both stages optimize the student with standard supervised fine-tuning (SFT) on teacher reasoning trajectories. LetDdenoteD 0 in Stage 1 orD distill t in Stage 2, where responses include full reasoning and final answers. The objective is the token-normalized cross-entropy loss: L SFT (θ;D) =− P (x i ,y i )∈D P |y i | j=1 logp θ (y ij | x i ,h i ,y i,<j ) P (x i ,y i )∈D |y i | .(10) Only response tokens contribute to the loss. Stage 1 obtainsM 0 through direct distillation, while each Stage 2 turn updates M t with verified capability-boundary trajectories to obtain M t+1 . 5 Preprint. Under Review. Table 1: Comparison with Reasoning Models under Different Scales and Training Budgets. Scores are percentages. Data denotes the number of publicly disclosed reasoning post-training examples and – indicates unavailable information. ModelParamsData Mathematical ReasoningGeneral Reasoning AIME AIMEHMMT HMMT BruMOIMOAMOArxivZebra GPQA Super 2526252625-Ans -Bench MathLogic-DGPQA Small reasoning models SmolLM3-3B3B–36.741.026.036.449.228.75.37.826.041.729.2 VibeThinker-3B3B–91.494.389.380.393.876.441.027.274.770.234.8 DASD-4B-Thinking4B448K83.388.380.072.785.061.626.59.769.068.442.9 Hunyuan-4B-Instruct4B–66.557.735.245.562.739.64.75.883.561.138.3 Qwen3-4B-Thinking-25074B–81.379.055.557.677.751.618.718.491.865.847.8 Qwen3.5-4B4B– 79.884.073.868.983.562.926.825.292.476.252.9 NVIDIA-Nemotron-3-Nano4B–78.5–53.2– Spark-X2.5-4B4B–90.090.7–81.2–74.2–60.567.4– Olmo-3-7B-Think7B >102K67.969.143.839.469.049.414.014.665.046.2– MiMo-7B-RL-05307B6M 70.276.048.554.679.853.914.011.770.960.6– OpenReasoning-Nemotron7B ≥5M78.280.263.551.578.860.626.015.519.861.1– Gemma-4-12B-it12B–72.977.563.372.780.454.917.333.092.178.8– Phi-4-reasoning-plus14B– 68.473.650.351.566.546.24.715.5–81.9– Ministral-3-Reasoning-251214B–82.985.067.142.486.763.420.719.485.171.2– Large reasoning models gpt-oss-20b (high)20B– 91.790.276.7–86.761.9–71.544.9 NVIDIA-Nemotron-3-Nano30B–89.189.984.6–70.4–74.1– Nemotron-3.5-Lightning30B–75.4– GLM-4.7-Flash30B–91.688.3–73.853.8–75.2– GLM-4.5-Air106B– 83.3–69.2–90.0–75.0– Qwen3-235B-A22B-Thinking 235B–92.3–83.9–70.547.8–97.581.160.7 LongCat-Flash-Thinking560B–90.6–83.7–43.6–95.581.5– GPT-5 nano (high)N/A– 85.276.775.669.780.8–19.491.271.248.9 Our method BoundaryEvo-4B4B1K92.593.390.082.395.875.346.035.996.081.355.0 4EXPERIMENTS 4.1EXPERIMENTAL SETUP Models and data. We evaluate BoundaryEvo primarily on Qwen3.5-4B (Qwen Team, 2026), with additional experiments on Qwen3.5-2B (Qwen Team, 2026) and Gemma-4-E4B-it (Team et al., 2026). All QA pairs are sampled from a cleaned subset (mathnotool.jsonl) of Nemotron-Cascade-2- SFT-Data (NVIDIA, 2026a), with reasoning trajectories re-distilled from DeepSeek-V3.2-Speciale (DeepSeek-AI et al., 2025). Stage 1 uses 100 teacher trajectories, while Stage 2 selects from 1,999 candidate questions. GPT-5.1 (OpenAI, 2025) provides expert rollouts for capability-boundary estimation, and Gemini 3.5 Flash (Google DeepMind, 2026) serves as the analyzer. Across three evolution turns, BoundaryEvo trains on 1,000 examples in total (100 + 3×300). Training and Evaluation. We train models with LoRA-based SFT (Hu et al., 2022). Each evolution turn continues from the previous checkpoint using the selected training trajectories. Training details, including optimization settings and implementation configurations, are provided in Appendix B. We evaluate reasoning performance on eight mathematical benchmarks, including AIME25, AIME26 (Mathematical Association of America, 2026), HMMT25 (HMMT, 2025), HMMT26 (HMMT, 2026), BruMO25 (BRUMO, 2025), IMO-AnswerBench (IMO-Ans) (Luong et al., 2025), AMO-Bench (An et al., 2025), and ArxivMath (Dekoninck et al., 2026). We further measure general reasoning transfer on ZebraLogicBench (ZebraLogic) (Lin et al., 2025), GPQA-Diamond (GPQA-D) (Rein et al., 2023), and SuperGPQA (Team et al., 2025c). All scores are reported as percentages, with Math and General denoting unweighted averages over the corresponding benchmark groups. Baselines. We compare zero-shot performance, Stage 1 priming, random selection, and LLM2LLM baseline (Lee et al., 2024). All evolution methods share the same Stage 1 checkpoint and training budget to isolate the effect of Stage 2 data acquisition. The LLM2LLM baseline selects new training examples based on student errors under the same training protocol. We additionally include a static 6 Preprint. Under Review. Table 2: Qwen3.5-4B results across three evolution turns. All Stage 2 strategies start from the same Stage 1 checkpoint. Scores are percentages, with Math and General averaged over eight and three benchmarks. Superscripts indicate absolute changes from zero-shot. Bold and underlining mark the highest and second-highest values in each column. Method Mathematical ReasoningGeneral ReasoningAvg. AIMEAIME HMMT HMMT BruMOIMOAMOArxivZebraGPQASuper MathGeneral 2526252625-Ans-BenchMathLogic-DGPQA Qwen3.5-4B + Zero-shot79.884.073.868.983.562.926.825.292.476.252.963.173.8 + Stage 184.2 +4.4 88.3 +4.3 81.7 +7.9 69.7 +0.8 85.0 +1.5 66.3 +3.4 38.5 +11.7 28.2 +3.0 92.5 +0.1 79.8 +3.6 53.4 +0.5 67.7 +4.6 75.2 +1.4 Stage 2 with Random Selection + Turn 185.0 +5.2 88.3 +4.3 82.4 +8.6 72.4 +3.5 80.8 -2.7 66.5 +3.6 38.5 +11.7 27.2 +2.0 92.7 +0.3 79.0 +2.8 52.8 -0.1 67.6 +4.5 74.8 +1.0 + Turn 285.3 +5.5 90.6 +6.6 82.9 +9.1 70.0 +1.1 86.7 +3.2 66.8 +3.9 38.7 +11.9 28.3 +3.1 93.0 +0.6 80.2 +4.0 52.9 +0.0 68.7 +5.6 75.4 +1.6 + Turn 384.0 +4.2 86.7 +2.7 85.6 +11.8 74.8 +5.9 79.7 -3.8 66.2 +3.3 39.0 +12.2 25.9 +0.7 93.9 +1.5 78.1 +1.9 52.8 -0.1 67.7 +4.6 74.9 +1.1 Stage 2 with LLM2LLM + Turn 186.0 +6.2 89.2 +5.2 82.1 +8.3 70.8 +1.9 84.8 +1.3 66.6 +3.7 39.1 +12.3 28.4 +3.2 93.3 +0.9 77.8 +1.6 53.1 +0.2 68.4 +5.3 74.7 +0.9 + Turn 286.7 +6.9 90.7 +6.7 83.3 +9.5 71.0 +2.1 84.2 +0.7 67.0 +4.1 38.9 +12.1 28.4 +3.2 94.0 +1.6 78.8 +2.6 53.2 +0.3 68.8 +5.7 75.3 +1.5 + Turn 390.7 +10.9 90.0 +6.0 84.3 +10.5 72.7 +3.8 86.7 +3.2 67.4 +4.5 39.8 +13.0 28.2 +3.0 95.6 +3.2 79.3 +3.1 53.8 +0.9 70.0 +6.9 76.2 +2.4 Stage 2 with BoundaryEvo (Ours) + Turn 1 89.3 +9.5 91.7 +7.7 86.0 +12.2 75.8 +6.9 85.0 +1.5 69.0 +6.1 41.5 +14.7 31.1 +5.9 92.8 +0.4 79.3 +3.1 54.3 +1.4 71.2 +8.1 75.5 +1.7 + Turn 290.0 +10.2 92.5 +8.5 86.7 +12.9 81.8 +12.9 90.0 +6.5 73.3 +10.4 44.0 +17.2 28.2 +3.0 95.9 +3.5 80.8 +4.6 54.1 +1.2 73.3 +10.2 76.9 +3.1 + Turn 392.5 +12.7 93.3 +9.3 90.0 +16.2 82.3 +13.4 95.8 +12.3 75.3 +12.4 46.0 +19.2 35.9 +10.7 96.0 +3.6 81.3 +5.1 55.0 +2.1 76.4 +13.3 77.4 +3.6 full-data SFT baseline trained on 1,000 examples for comparison with fixed data training. Besides, Table 1 further compares our final model with representative reasoning models across different parameter scales. The small- and medium-scale group includes SmolLM3-3B (Bakouch et al., 2025), VibeThinker-3B (Xu et al., 2026), DASD-4B-Thinking (Yan et al., 2026), Hunyuan-4B-Instruct (Tencent Hunyuan Team, 2025), Qwen3-4B-Thinking-2507 (Qwen Team, 2025), Qwen3.5-4B (Qwen Team, 2026), NVIDIA-Nemotron-3-Nano-4B-BF16 (NVIDIA, 2026c), Spark-X2.5-4B (Team, 2026), Olmo-3-7B-Think (Olmo et al., 2026), MiMo-7B-RL-0530 (Xiaomi et al., 2025), OpenReasoning- Nemotron-7B (NVIDIA, 2025), Gemma-4-12B-it (Team et al., 2026), Phi-4-reasoning-plus (Abdin et al., 2025), and Ministral-3-14B-Reasoning-2512 (Liu et al., 2026). The large-scale group includes gpt-oss-20b (OpenAI et al., 2025), NVIDIA-Nemotron-3-Nano-30B-A3B-BF-16 (NVIDIA et al., 2025), NVIDIA-Nemotron-3.5-Lightning-30B-A3B-BF16 (NVIDIA, 2026b), GLM-4.7-Flash (Team et al., 2025a), GLM-4.5-Air (Team et al., 2025a), Qwen3-235B-A22B-Thinking (Yang et al., 2025), LongCat-Flash-Thinking (Team et al., 2025b), and GPT-5 nano (Singh et al., 2026). 4.2DOWNSTREAM EFFECTIVENESS BoundaryEvo-4B achieves strong reasoning performance with only 1K training examples across diverse mathematical benchmarks. As summarized in Table 1, it obtains leading results on AIME25, HMMT25, HMMT26, BruMO25, and ArxivMath, and consistently outperforms other 4B-scale rea- soning models. Compared with DASD-4B-Thinking, BoundaryEvo improves all shared benchmarks, including a 19.5-point gain on AMO-Bench, while using only a small fraction of the training data. Beyond same-scale comparisons, BoundaryEvo remains competitive with much larger reasoning models, achieving 75.3 on IMO-AnswerBench and outperforming Qwen3-235B-A22B-Thinking. It also generalizes to broader reasoning tasks, reaching 96.0 on ZebraLogicBench, 81.3 on GPQA- Diamond, and 55.0 on SuperGPQA. These results demonstrate that capability-boundary-guided data–model co-evolution enables small reasoning models to effectively extract high-value long-chain reasoning supervision under limited data budgets. 4.3MAIN RESULTS As shown in table 2, BoundaryEvo achieves consistent gains through capability priming and subse- quent evolution. Stage 1 improves the mathematical average from 63.1 to 67.7, and three evolution turns further increase it to 71.2, 73.3, and 76.4, yielding an 8.7-point gain over priming. The general-reasoning average also improves from 75.2 to 77.4 after evolution, despite training only 7 Preprint. Under Review. Table 3:Transfer across model size and architecture. Each backbone is primed before three sequential Stage 2 turns. Scores are percentages. Superscripts indicate absolute changes from zero- shot. Bold/underline denote the top two values, and blue shading marks the best checkpoint. Method Mathematical ReasoningGeneral ReasoningAvg. AIMEAIMEHMMTHMMTBruMOIMOZebraGPQA MathGeneral 2526252625-AnsLogic-D Qwen3.5-2B + Zero-shot34.238.822.922.735.022.044.950.029.347.5 + Stage 138.3 +4.1 50.0 +11.2 26.7 +3.8 24.5 +1.8 43.3 +8.3 35.2 +13.2 50.5 +5.6 50.5 +0.5 36.3 +7.0 50.5 +3.0 + Turn 1 40.0 +5.8 46.7 +7.9 30.0 +7.1 33.3 +10.6 46.7 +11.7 35.7 +13.7 63.7 +18.8 52.5 +2.5 38.7 +9.4 58.1 +10.6 + Turn 246.7 +12.5 56.7 +17.9 33.3 +10.4 27.3 +4.6 51.7 +16.7 35.9 +13.9 68.7 +23.8 53.5 +3.5 41.9 +12.6 61.1 +13.6 + Turn 3 43.3 +9.1 46.7 +7.9 23.3 +0.4 30.0 +7.3 54.2 +19.2 35.2 +13.2 63.8 +18.9 51.5 +1.5 38.8 +9.5 57.7 +10.2 Gemma-4-E4B-it + Zero-shot36.753.326.739.450.831.669.458.639.864.0 + Stage 170.0 +33.3 76.7 +23.4 50.0 +23.3 42.4 +3.0 55.4 +4.6 44.9 +13.3 90.1 +20.7 60.1 +1.5 56.6 +16.8 75.1 +11.1 + Turn 1 76.7 +40.0 83.3 +30.0 80.0 +53.3 33.3 -6.1 55.7 +4.9 49.5 +17.9 92.8 +23.4 60.6 +2.0 63.1 +23.3 76.7 +12.7 + Turn 276.7 +40.0 86.7 +33.4 66.7 +40.0 60.6 +21.2 60.8 +10.0 50.3 +18.7 94.1 +24.7 61.1 +2.5 67.0 +27.2 77.6 +13.6 + Turn 380.0 +43.3 86.7 +33.4 70.0 +43.3 66.7 +27.3 67.5 +16.7 51.3 +19.7 94.7 +25.3 66.2 +7.6 70.4 +30.6 80.5 +16.5 on mathematical data. BoundaryEvo outperforms random selection and LLM2LLM on all eleven benchmarks, exceeding them by 8.7 and 6.4 points on mathematical averages and by 2.5 and 1.2 points on general reasoning. The largest gains over LLM2LLM appear on HMMT26, BruMO25, IMO-AnswerBench, and ArxivMath, highlighting the advantage of capability-boundary-driven acqui- sition over error-based selection. Across evolution turns, random selection quickly saturates, while LLM2LLM improves more slowly; in contrast, BoundaryEvo consistently improves across both benchmark groups. These results show that capability-boundary-driven selection enables progressive improvement through iterative QA selection and verified trajectory distillation. 4.4GENERALIZATION ACROSS MODEL BACKBONES Table 3 evaluates the transferability of BoundaryEvo across model scales and architectures. For Qwen3.5-2B, Stage 1 improves the mathematical and general-reasoning averages from 29.3 to 36.3 and from 47.5 to 50.5, respectively. Subsequent boundary-guided evolution improves performance, with Turn 2 achieving the strongest checkpoint (e.g., AIME25: 46.7, AIME26: 56.7, and ZebraLog- icBench: 68.7). However, the smaller model shows earlier saturation and less stable gains across turns, suggesting that limited model capacity can constrain long-term evolution. For Gemma-4-E4B-it, BoundaryEvo achieves more consistent improvements: Stage 1 raises AIME-25 from 36.7 to 70.0 and ZebraLogicBench from 69.4 to 90.1, while Stage 2 further improves the mathematical and general-reasoning averages to 70.4 and 80.5 at Turn 3. These results demonstrate that BoundaryEvo transfers across different backbones and adapts its evolution process to varying student capabilities. 4.5TRAINING DYNAMICS The training dynamics reveal the progressive adaptation enabled by BoundaryEvo. As illustrated in Figure 3(a), fitting the selected teacher trajectories across three evolution turns consistently reduces the mean loss from 0.4362 to 0.3936 and 0.3388, indicating stable optimization on newly discovered boundary examples with replay-based training. As shown in Figure 3(b), the selected curriculum gradually shifts as the student improves. The mean student score increases from 0.233 to 0.265 and 0.271 while expert scores remain high at 0.846, 0.879, and 0.884, preserving a meaningful student– expert gap for continued selection of challenging samples. Together, these trends demonstrate capability-boundary-guided co-evolution, where the curriculum evolves with student capability while maintaining reliable supervision and continuously expanding the training frontier. 4.6DISCUSSION Does continued evolution improve the training frontier? As shown in Figure 4, BoundaryEvo progressively improves IMO-AnswerBench from 66.3 after Stage 1 to 69.0, 73.3, and 75.3 across three evolution turns. The final checkpoint surpasses Stage 1 and static full-data SFT by 9.0 and 5.8 points, respectively. These gains show that continuously updating the training distribution according to the student capability boundary is more effective than using a fixed dataset. 8 Preprint. Under Review. 255075100125150175 Training Steps 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 SFT loss Turn 1Turn 2Turn 3 Turn 1 Turn 2 Turn 3 (a) 0.80 0.85 0.90 0.846 0.879 0.884 Student ModelExpert Model 0.6130.6140.613 Turn 1Turn 2Turn 3 Evolution Turns 0.20 0.25 0.30 0.233 0.265 0.271 Analyzer Evaluation Score (b) Figure 3: Training and selection dynamics across the Stage 2. (a) SFT loss for representative runs. (b) Mean student and expert scores on each turn. Annotations show the score gap. Base Stage 1 Turn 1Turn 2Turn 3 Evolution Stages 60.0 62.5 65.0 67.5 70.0 72.5 75.0 77.5 Evaluation Score (%) Priming StageCo-evolution Stage Ours (Stage 1 + Stage 2) Ours w/o Stage 2 Replay Ours w/ Stage 1 Verification Zero-shot Full-data SFT 62.9 66.3 69.0 73.3 75.3 66.3 68.8 68.2 71.1 64.1 66.9 70.4 69.9 62.9 69.5 Figure 4: Ablation study on IMO-AnswerBench performance. The curves compare BoundaryEvo with verified Stage 1 and no-replay variants, while reference lines show zero-shot and static SFT on 1K examples. Does capability priming benefit from judge- free distillation? Table 4 shows that remov- ing Stage 1 decreases the final mathematical and general reasoning averages from 76.4/77.4 to 73.5/74.0, demonstrating the importance of capa- bility priming before evolution. Moreover, verified Stage 1 further reduces performance to 73.1/73.9. As illustrated in Figure 4, direct distillation starts higher and maintains a larger advantage through- out evolution. These results suggest that Stage 1 benefits from preserving diverse teacher reasoning behaviors, while verification is better suited for Stage 2 boundary refinement. Does replay stabilize co-evolution? Removing replay reduces IMO-AnswerBench from 75.3 to 71.1 at the final turn. Although newly selected boundary examples improve the student, the per- formance drop indicates that learning only from the latest curriculum can gradually weaken pre- viously acquired capabilities. Replay therefore helps retain historical knowledge while adapting to newly discovered boundary regions. Table 4: Ablation study on effect of Stage 1 on final Qwen3.5-4B performance. Math and Gen- eral average eight and three benchmarks. MethodMath AVGGeneral AVG BoundaryEvo76.477.4 ⊢ Stage 1 (w/ verification)73.173.9 ⊢ w/o Stage 173.574.0 Overall, the ablations confirm the complementary roles of BoundaryEvo components: Stage 1 estab- lishes the initial capability foundation, boundary- guided evolution expands the training frontier, and replay stabilizes continual improvement. 5CONCLUSION We introduce BoundaryEvo, a dynamic capability-boundary co-evolution framework for long-chain reasoning distillation. Unlike conventional methods that treat reasoning data as a static resource, BoundaryEvo continuously adapts training supervision according to the evolving capabilities of the student model. It first performs judge-free capability priming to acquire diverse teacher reasoning behaviors, followed by capability-boundary-guided data selection and verified teacher trajectory distillation to discover progressively valuable reasoning supervision. Experiments across model backbones demonstrate that BoundaryEvo achieves strong reasoning performance with only 1K training examples, while ablations validate the importance of capability priming and capability- boundary-guided evolution. These results show that effective reasoning supervision is not fixed, but can be continuously discovered through data–model co-evolution. 9 Preprint. Under Review. AI USE STATEMENT We used generative AI tools for limited assistance in language refinement and writing improvement. We did not use generative AI tools for generating research ideas, designing experiments, conducting evaluations, or producing scientific claims. All AI-assisted content was reviewed and revised by the authors, and the authors take full responsibility for the final content of this work. ETHICS STATEMENT This study concerns the training and evaluation of reasoning models. Training questions are derived from a subset of the publicly released Nemotron-Cascade-2-SFT-Data. Evaluation uses eleven publicly released benchmarks: AIME 2025, AIME 2026, HMMT 2025, HMMT 2026, BruMO 2025, IMO-AnswerBench, AMO-Bench, ArxivMath, ZebraLogicBench, GPQA-Diamond, and SuperGPQA. These datasets and benchmarks were obtained from their public sources and used subject to their respective licenses and terms of use. This work does not recruit human participants or collect private user data, and we do not intentionally introduce personally identifiable information into the training or evaluation data. The empirical comparison reports results for 22 external systems in addition to BoundaryEvo: SmolLM3-3B, VibeThinker-3B, DASD-4B-Thinking, Hunyuan-4B- Instruct, Qwen3-4B-Thinking-2507, Qwen3.5-4B, NVIDIA-Nemotron-3-Nano-4B-BF16, Spark- X2.5-4B, Olmo-3-7B-Think, MiMo-7B-RL-0530, OpenReasoning-Nemotron-7B, Gemma-4-12B-it, Phi-4-reasoning-plus, Ministral-3-14B-Reasoning-2512, gpt-oss-20b, NVIDIA-Nemotron-3-Nano- 30B-A3B-BF16, NVIDIA-Nemotron-3.5-Lightning-30B-A3B-BF16, GLM-4.7-Flash, GLM-4.5- Air, Qwen3-235B-A22B-Thinking, LongCat-Flash-Thinking, and GPT-5 nano. Cross-backbone experiments additionally use Qwen3.5-2B and Gemma-4-E4B-it. Data construction uses DeepSeek- V3.2-Speciale as the trajectory teacher, GPT-5.1 as the capability-boundary expert, and Gemini 3.5 Flash as the analyzer. Open checkpoints were accessed from their released sources, while proprietary models were accessed through authorized APIs under the providers’ terms of service. REPRODUCIBILITY STATEMENT We provide detailed descriptions of the BoundaryEvo framework, data construction process, training procedure, and evaluation settings in the main paper and supplementary materials. Implementation details, including optimization configurations and hyperparameters, are provided in the appendix. We will release the code and processed data to facilitate reproduction and future research. REFERENCES Marah Abdin, Sahaj Agarwal, Ahmed Awadallah, Vidhisha Balachandran, Harkirat Behl, Lingjiao Chen, Gustavo de Rosa, Suriya Gunasekar, Mojan Javaheripi, Neel Joshi, Piero Kauffmann, Yash Lara, Caio C ́ esar Teodoro Mendes, Arindam Mitra, Besmira Nushi, Dimitris Papailiopoulos, Olli Saarikivi, Shital Shah, Vaishnavi Shrivastava, Vibhav Vineet, Yue Wu, Safoora Yousefi, and Guoqing Zheng. Phi-4-reasoning technical report, 2025. URLhttps://arxiv.org/abs/ 2504.21318. 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Stage 2 data count is300Rand counts replayed examples whenever they are reused. Each non-reference row changes one factor only. Ablated factor E R τ g Stage 2 dataMath / General Reference530.2090076.4 / 77.4 Epochs130.2090072.6 / 75.9 Epochs330.2090073.5 / 76.1 Epochs1030.2090074.3 / 77.0 Evolution turns510.2030071.2 / 75.5 Evolution turns520.2060073.3 / 76.9 Evolution turns540.20120075.8 / 77.0 Minimum gap530.1090076.1 / 77.2 Minimum gap530.3090076.2 / 77.2 The epoch sweep tests whether the result depends on under- or over-training the selected trajectories. The turn sweep measures the benefit of repeatedly relocating the capability boundary as the student changes. The gap sweep tests whether selection is robust to admitting easier examples or requiring a stricter separation between expert and student performance. Together, these compact controls isolate the most consequential optimization, evolution, and selection choices. BTRAINING DETAILS This section reports the optimization and implementation details used for capability priming and the subsequent evolution turns. Unless stated otherwise, Stage 1 and Stage 2 use the same supervised fine-tuning recipe. The principal differences across backbones are the conversation template, the LoRA target modules, and long-sequence preprocessing. Optimization and Parameter-Efficient Fine-Tuning We train all student models with response-only LoRA SFT using LLaMA-Factory. Tokens belonging to the question and system prompt are masked from the loss, so optimization is applied only to the teacher response. For Qwen3.5, LoRA is attached to all linear modules. For Gemma-4-E4B-it, it is attached to the query, key, value, output, gate, up, and down projection modules. Table 6 summarizes the remaining settings. Training jobs use one or four H200 GPUs depending on resource availability while preserving the default effective batch size of eight. We use gradient accumulation of 2 for four-GPU jobs and 8 for one-GPU jobs. With 100 Stage 1 examples, five epochs correspond to 65 optimizer updates. A standard 300-example Stage 2 run contains 190 updates. Stage-Wise Initialization and Training Budget Stage 1 starts from the original pretrained backbone and trains on 100 direct teacher trajectories. Its merged checkpoint is denoted byM 0 . Stage 2 is sequential: Turn 1 starts fromM 0 , and Turnt + 1starts from the merged student produced by Turn t. Each turn contains 300 verified teacher trajectories selected from the corresponding capability- boundary question set. Thus, the reported three-turn configuration allocates100 + 3× 300 = 1000 training examples. This accounting counts an example again when it is replayed in a later turn. After each update, we merge the learned LoRA adapter into its initialization checkpoint before using the resulting model for evaluation or as the next turn’s student. Consequently, every evolution turn 21 Preprint. Under Review. Table 6: Training configuration used in experiments. The effective batch size is the product of the per-device batch size, gradient accumulation steps, and the number of data-parallel workers. SettingQwen3.5-4B / 2BGemma-4-E4B-it Training objectiveResponse-only SFTResponse-only SFT Fine-tuning methodLoRALoRA LoRA rank / scaling factor64 / 12864 / 128 LoRA target modulesAll linear modulesq,k,v,o , gate, up, and down pro- jections Trainable parameters129.9M / 67.3M139.5M Conversation template qwen3 gemma4 Epochs per update55 Optimizer AdamW, fused PyTorch imple- mentation Same Learning rate1× 10 −4 1× 10 −4 Scheduler / warm-upCosine / 10%Cosine / 10% Adam β 1 ,β 2 ,ε0.9, 0.999, 10 −8 Same Weight decay / gradient clipping 0 / 1.0Same Numerical precisionbfloat16bfloat16 Distributed optimizationDeepSpeed ZeRO-3DeepSpeed ZeRO-3 Gradient checkpointingEnabledEnabled Per-device batch size11 Default effective batch size88 Maximum trainer sequence length 40,960 tokens40,960 tokens Maximum retained training length 40,960 tokens32,768 tokens Random seed4242 selects data with the fully updated student rather than with an unmerged adapter or the original base model. Random-selection and LLM2LLM baselines use the same SFT objective, optimizer family, epoch count, and backbone-specific formatting. Long-Trajectory Preprocessing Training records are stored in OpenAI-stylemessagesformat and contain only fields consumed by the trainer. Qwen3.5 runs use theqwen3template and a 40,960-token cutoff. For Gemma-4-E4B-it, we render each record with the nativegemma4template and tokenizer before training. We retain only examples whose rendered length does not exceed 32,768 tokens, while keeping the trainer ceiling at 40,960 tokens. This separates data selection from implicit truncation and ensures that accepted Gemma trajectories remain complete. Reasoning markers are converted to the backbone’s native chat representation before tokenization. Across backbones, malformed records and examples without a usable assistant response are discarded during preprocessing rather than padded with synthetic content. CREASONING-TRAJECTORY LENGTH ANALYSIS We test whether the accuracy gains in Table 2 require uniformly longer generations. We compare Qwen3.5-4B and the final BoundaryEvo-4B checkpoint on seven benchmarks, retaining only exact normalized question matches. Repeated rollouts are summarized by their per-question median, so every matched item has equal weight. We measure reasoning tokens with the Qwen tokenizer after removing the final-answer segment and three thinking-boundary tokens. The final model has a lower median reasoning length on every matched benchmark, with reductions ranging from 7.4% on ArxivMath to 52.9% on AIME26. These reductions coincide with higher accuracy on all seven tasks in Table 2. For example, AIME26 improves from 84.0 to 93.3 while its median trajectory falls from 36.9K to 17.4K tokens, and ZebraLogic improves from 92.4 to 96.0 22 Preprint. Under Review. 020406080100 0.00 0.01 0.02 0.03 0.04 Density AIME25 (n= 30) 020406080100 0.000 0.005 0.010 0.015 0.020 0.025 0.030 AIME26 (n= 30) 020406080100 0.000 0.005 0.010 0.015 0.020 0.025 0.030 HMMT25 (n= 30) 020406080 Reasoning tokens (thousands) 0.000 0.005 0.010 0.015 0.020 0.025 HMMT26 (n= 31) 020406080 Reasoning tokens (thousands) 0.000 0.005 0.010 0.015 0.020 Density ArxivMath (n= 103) 01020304050 Reasoning tokens (thousands) 0.00 0.02 0.04 0.06 GPQA-D (n= 186) 0204060 Reasoning tokens (thousands) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 ZebraLogic (n= 1000) −50−40−30−20−100 Median change (%) AIME25 AIME26 HMMT25 HMMT26 Arxiv GPQA-D Zebra -52% -53% -41% -24% -7% -25% -48% Paired length change Qwen3.5-4BBoundaryEvo-4B Figure 5: Reasoning-length distributions on matched evaluation questions. Curves show smoothed densities of per-question reasoning-token counts, and vertical lines mark medians. The final panel reports the relative change in median length from Qwen3.5-4B to BoundaryEvo-4B. Repeated rollouts of the same question are summarized by their median before plotting. Table 7: Reasoning-length statistics for the matched questions in Figure 5. Token counts are reported in thousands.∆denotes the relative change in median length from the base model to BoundaryEvo- 4B. BenchmarknMedian∆90th percentile BaseOursBaseOurs AIME253042.820.5−52.2%99.452.1 AIME263036.917.4−52.9%74.954.9 HMMT253046.127.3−40.8%102.474.3 HMMT263145.434.6−23.9%71.374.6 ArxivMath10356.452.2−7.4%82.171.2 GPQA-D1869.57.1−25.2%17.436.4 ZebraLogic10007.33.8−47.6%25.716.5 while its median falls from 7.3K to 3.8K tokens. The result indicates that the gains do not arise from a uniform increase in inference length. Upper-tail behavior remains task dependent. The 90th percentile falls on five tasks, changes little on HMMT26, and rises on GPQA-D. Thus, BoundaryEvo does not simply truncate every response. GPQA-D combines a lower median with a longer tail. Although this observational analysis does not establish causality, the joint accuracy gains and lower typical lengths are consistent with more efficient, rather than merely more verbose, reasoning. DGPQA-D SCALING AND DATA EFFICIENCY We further examine how BoundaryEvo compares with models of different scales and reported post- training budgets. Both plots in this section use GPQA-D as the common performance measure and take their values directly from Table 1. We omit systems whose parameter count or GPQA-D result is unavailable rather than estimating missing values. Figure 6 shows that strong GPQA-D performance is not determined by parameter count alone. BoundaryEvo-4B reaches 81.3, improving over its Qwen3.5-4B base model by 5.1 points and exceeding every reported model up to 12B parameters. It is also competitive with substantially larger systems, including Qwen3-235B at 81.1 and LongCat-560B at 81.5, while Phi-4-Reasoning at 14B obtains the highest displayed score of 81.9. These results place BoundaryEvo on the upper performance frontier of the plot, rather than suggesting that it dominates every larger model. 23 Preprint. Under Review. 10 1 10 2 Model Parameters (B, log scale) 40 50 60 70 80 GPQA-D Accuracy (%) GPQA-D Performance versus Model Scale Bubble area: parameters 4B 30B 235B SmolLM3 VibeThinker-3B DASD-4B Hunyuan-4B Qwen3-4B Qwen3.5-4B Nemotron-4B Spark-X2.5-4B Olmo-3-Think MiMo-7B OR-Nemotron-7B Gemma-4-it Phi-4-Reasoning Ministral-3 GPT-OSS-20B Nemotron-3-Nano Nemotron-3.5 GLM-4.7-Flash GLM-4.5-Air Qwen3-235B LongCat-560B BoundaryEvo-4B Compact (3B--4B)Mid-scale (7B--14B)Large (20B--106B)Frontier (235B+)BoundaryEvo-4B Figure 6: GPQA-D performance versus model scale. The horizontal axis is logarithmic, bubble area encodes parameter count, and color denotes the parameter range. BoundaryEvo-4B is highlighted with a white star. Only models with both a reported parameter count and GPQA-D score in Table 1 are included. 10 3 10 4 10 5 10 6 10 7 Reported Post-training Examples (log scale) 45 50 55 60 65 70 75 80 85 GPQA-D Accuracy (%) GPQA-D Performance versus Training Data BoundaryEvo-4B Comparison model Reported lower bound Bubble area: parameters 4B7B BoundaryEvo-4B Olmo-3-Think (reported lower bound) DASD-4B OR-Nemotron-7B (reported lower bound) MiMo-7B Figure 7: GPQA-D performance versus publicly reported reasoning post-training data. The horizontal axis is logarithmic and bubble area encodes model parameters. Right-pointing arrows mark reported lower bounds rather than exact dataset sizes. Only systems with disclosed data amounts and GPQA-D scores in Table 1 are shown. The data-budget view in Figure 7 highlights a larger separation. BoundaryEvo uses 1K cumulative training examples and scores 81.3. DASD-4B uses 448K reported examples and scores 68.4, while the three 7B comparison models report at least 102K to 6M examples and score from 46.2 to 61.1. Thus, within this reported set, BoundaryEvo achieves the best GPQA-D result with the smallest disclosed training set. 24 Preprint. Under Review. This comparison measures example efficiency, not total computational cost. Dataset definitions, trajectory lengths, and filtering policies differ across papers. Moreover, BoundaryEvo’s 1K count includes repeated training exposure but excludes student and expert inference, analyzer judging, and teacher-rollout generation. The plots should therefore be read as a compact view of reported model and data scale, rather than as a controlled comparison of training FLOPs. EBOUNDARYEVO ALGORITHM AND IMPLEMENTATION DETAILS End-to-End Training Procedure. Algorithm 1 summarizes the two-stage pipeline, where Stage 1 establishes reasoning capability through judge-free distillation and Stage 2 evolves supervision by selecting capability-boundary questions for verified teacher trajectory distillation. Algorithm 1: BoundaryEvo: Capability Priming and Capability-Boundary Co-Evolution Input: Base student M base ; trajectory teacher T ; fixed expert E; analyzer J ; priming questions Q 0 ; evolution poolP ; reference answersy ref i ; thresholdsτ ; number of turns R; QA budget K = 300; teacher rollout count L Output: Evolved student M R // Stage 1: learn how the teacher reasons D 0 ←∅; foreach x i ∈Q 0 do Sample a holistic trajectory y T i ∼ T (·| x i ); if y T i passes basic response-validity checks then D 0 ←D 0 ∪(x i ,y T i ); M 0 ← SFT(M base ,D 0 ); D QA −1 ←∅; // Stage 2: learn what the student should learn next for t← 0 to R− 1 do A t ←∅; foreach z i = (x i ,h i ,y ref i )∈P do Independently sample ˆy s i,t ∼ M t (·| x i ,h i ) and ˆy e i ∼ E(·| x i ,h i ); a (t) i ← J (z i , ˆy s i,t , ˆy e i ); Compute g (t) i = s e i − s s,(t) i ; if s e i ≥ τ e , s s,(t) i < τ s , g (t) i ≥ τ g , q i ≥ τ q , b clear i = b answerable i = 1, and c i ≥ τ c then A t ←A t ∪z i ; R t ←z ∈A t : x(z)∈ Questions(D QA t−1 ); N t ←A t t ; Retain eligible replay examples up to the turn-specific replay quota; Rank novel examples by student-rollout completeness, analyzer recommendation, and r (t) i ; Fill the remaining slots with ranked novel examples to formD QA t , where|D QA t | = K; // Construct verified supervision only after QA selection D distill t ←∅; foreach z i ∈D QA t do Sampley T,(k) i,t L k=1 independently from T (·| x i ,h i ); Y + i,t ←y : V (Ans(y),y ref i ) = 1; ifY + i,t ̸=∅ then Select y ⋆ i,t ∈Y + i,t and add (x i ,h i ,y ⋆ i,t ) toD distill t ; M t+1 ← SFT(M t ,D distill t ); return M R ; Replay and Training-Set Construction. At turnt, accepted questions are divided by question identity relative to the immediately preceding training set. Replay candidatesR t were used in the 25 Preprint. Under Review. preceding turn and remain eligible under the current screening protocol. Novel candidatesN t do not occur in that preceding set. The training budget remains fixed: D QA t = Select K R t (R t )∪ Select K N t (N t ), K R t + K N t = 300.(11) Eligible replay questions are retained up to the turn-specific replay quota. FPROMPT TEMPLATES We report the raw prompts used by the training-data pipeline. Placeholders in braces are replaced at runtime. Stage 1 has no model-based judge prompt. F.1STAGE 1 TEACHER TRAJECTORY DISTILLATION PROMPT The Stage 1 teacher receives the following system and user messages. The resulting trajectory is used directly after basic response-validity checks. Stage 1 Teacher Trajectory Distillation Prompt [System] You are a helpful and harmless assistant. You are not allowed to use any tools. [User] Please reason step by step, and put your final answer within . question F.2STAGE 2 STUDENT AND EXPERT ROLLOUT PROMPT The current student and fixed expert receive the same prompt and answer the question independently. Stage 2 Student and Expert Rollout Prompt Please solve the following problem independently. Show concise reasoning, then end with exactly one final-answer line: Final answer: <your answer> optional_context Problem: question F.3CAPABILITY-BOUNDARY ANALYZER PROMPT The analyzer receives the question, optional context, reference answer, and both independently generated responses. In the raw prompt, weak denotes the current student and strong denotes the fixed expert. The implementation also records reference-correctness and leakage judgments as auxiliary diagnostics. Their hard rejection switches are disabled in the reported route. Capability-Boundary Analyzer Prompt [System] 26 Preprint. Under Review. You must output exactly one valid JSON object. Do not include markdown fences, comments, natural-language prefaces, or trailing text. [User] You are a strict data-quality and model-trajectory judge. Evaluate whether this example is suitable training data. Consider: - whether the question is clear and answerable; - whether the reference answer is correct; - if context is provided, whether the answer is supported by the context; - how good the weak-model answer is; - how good the strong-model answer is; - whether the strong answer is meaningfully better than the weak answer. Scores must be floats in [0, 1]. - weak_score: quality/correctness of the weak answer. - strong_score: quality/correctness of the strong answer. - data_quality_score: clarity, answerability, reference correctness, and training usefulness. - context_support_score: use 1.0 when no external context is required or no context is provided. Thresholds used by the pipeline: "min_strong_score": 0.65, "max_weak_score": 0.50, "min_gap": 0.20, "min_quality_score": 0.70, "min_context_support_score": 0.70 Question: question Context: context_or_no_context_provided Reference answer: reference_answer Weak model answer: student_answer Strong model answer: expert_answer Return exactly one JSON object with this schema: "is_clear": true, "is_answerable": true, "is_reference_correct": true, "is_context_supported": true, "weak_score": 0.25, "strong_score": 0.85, "data_quality_score": 0.82, "context_support_score": 0.9, "accepted_recommendation": true, "rejection_reasons": [], "rationale": "Strong model solves the example correctly, while weak model misses the key reasoning step." 27 Preprint. Under Review. F.4FINAL-ANSWER VERIFICATION PROMPT Each sampled teacher trajectory is judged by final-answer equivalence. Equivalent algebraic forms and reasonable numerical precision are accepted. Final-Answer Verification Prompt [System] You must output exactly one valid JSON object. Do not include markdown fences, comments, natural-language prefaces, or trailing text. [User] [JUDGE] You are a strict math/reasoning answer judge. Decide whether the candidate answer is mathematically equivalent to the reference answer. Allow equivalent forms, algebraic rearrangements, and reasonable numeric precision. Problem: question Reference answer: reference_answer Candidate answer: candidate_teacher_answer Return exactly one JSON object: "correct": true, "reason": "one concise sentence" GDESCRIPTION OF EXPERIMENTAL EVALUATION G.1COMMON EVALUATION PROTOCOL Provenance of reported scores. The values in Tables 1, 2, and 3 follow a documented source hierarchy. For model–benchmark pairs reported in the VibeThinker study, official model cards, or official benchmark leaderboards, we use the corresponding publicly reported scores. When a model or benchmark combination is not available from these sources, we perform a local evaluation, primarily for smaller open-weight models, using the EvalScope framework and the official evaluation split. These local runs retain EvalScope’s default top-pand temperature and a maximum of 102,400 generated tokens setting. The same provenance rule is applied across all three tables. Protocol for controlled follow-up experiments. All follow-up methods use the same official evaluation splits, prompts, answer extractors, decoding settings, evaluator versions, and generation seed schedule. Competition-math benchmarks use at least four independent completions per question, while all other benchmarks use four. We report the mean correctness of individual completions rather than majority vote, best-of-n, or pass@n. Unparseable and truncated responses without an extractable final answer are counted as incorrect rather than discarded. Math and General are the same unweighted benchmark averages used in the main paper, and the final checkpoint after three turns is fixed as the primary endpoint. All methods within a seed share the priming checkpoint, candidate pool, cached expert outputs, and available teacher samples. Before training, exact and near-duplicate questions between the priming/evolution data and every evaluation set are identified and manually checked; 28 Preprint. Under Review. confirmed overlaps are removed from the common training pool. We will release question-ID manifests, exclusion counts, and the complete evaluation configuration. G.2RELATIONSHIP TO LLM2LLM AND AGENTFRONTIER LLM2LLM (Lee et al., 2024) first identifies student errors and then asks a teacher to synthesize new examples targeting those errors. Our current fixed-pool baseline instead rescans the same candidate pool and selects questions answered incorrectly by the current student. It isolates error-driven acquisition under the same pool and training budget, but it is an LLM2LLM-inspired adaptation rather than a complete reproduction. We additionally implement LLM2LLM-style synthesis, which retains the original error-to-new-question step, independently verifies and de-duplicates the generated QA pairs, and uses the same 300-example budget per turn. The source revision, prompts, rejected generations, and all adaptations will be reported. AgentFrontier (Chen et al., 2025) also targets tasks that a weaker model cannot solve but a stronger, tool-augmented system can. BoundaryEvo focuses on a more specific setting: tool-free long-trajectory distillation for compact students, a fixed QA pool and expert, and repeated boundary re-estimation after each student update. Our claimed contribution is therefore the two-stage priming-and-evolution procedure and its behavior under a small supervision budget, rather than the general idea of ZPD-style filtering. 29

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