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Allspark: Weak to Strong Transfer via Alternating Chain of Thought
Kaizhao Liang, Junxiong Wang, Chen Liang, Zhendong Wang, Qiang Liu
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Abstract:Recent progress in frontier models has renewed interest in large-scale reinforcement learning (RL), but the cost of generating large-model rollouts makes even testing RL recipes expensive. We ask whether reasoning improvements learned by a small, weak model can benefit a larger, stronger model without using the strong model's rollouts during training. We introduce Allspark, a training and inference framework for weak-to-strong transfer through alternating chains of thought. A weak teacher is trained alongside a frozen copy of the same model; the two alternate reasoning segments, and the frozen model produces the final answer. At inference time, a stronger student replaces the frozen training partner, while both models remain fixed. Because they communicate through text, the teacher can steer students from different model families and with different tokenizers. We study Allspark at two scales: controlled Qwen experiments across math and reasoning, and larger-scale Inkling experiments on ARC-AGI-2. The Inkling experiments show accuracy gains in within-family and cross-family settings, including transfer to Kimi and Nemotron, with benefits that vary across inference settings. These findings motivate reusing a trained weak teacher across strong students and examining the resulting accuracy--token tradeoff.
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Preprint ALLSPARK: WEAK TO STRONG TRANSFER VIA ALTERNATING CHAIN OF THOUGHT Kaizhao Liang UT Austin kaizhaol@utexas.edu Junxiong Wang Together AI junxiong@cs.cornell.edu Chen Liang Microsoft chenliang1@microsoft.com Zhendong Wang Microsoft zhendong.wang@utexas.edu Qiang Liu UT Austin lqiang@cs.utexas.edu ABSTRACT Recent progress in frontier models has renewed interest in large-scale reinforcement learning (RL), but the cost of generating large-model rollouts makes even testing RL recipes expensive. We ask whether reasoning improvements learned by a small, weak model can benefit a larger, stronger model without using the strong model’s rollouts during training. We introduce Allspark, a training and inference framework for weak-to-strong transfer through alternating chains of thought. A weak teacher is trained alongside a frozen copy of the same model; the two alternate reasoning segments, and the frozen model produces the final answer. At inference time, a stronger student replaces the frozen training partner, while both models remain fixed. Because they communicate through text, the teacher can steer students from different model families and with different tokenizers. We study Allspark at two scales: controlled Qwen experiments across math and reasoning, and larger-scale Inkling experiments on ARC-AGI-2. The Inkling experiments show accuracy gains in within-family and cross-family settings, including transfer to Kimi and Nemotron, with benefits that vary across inference settings. These findings motivate reusing a trained weak teacher across strong students and examining the resulting accuracy–token tradeoff. 1INTRODUCTION 弟子不必不如师，师不必贤于弟子。 A student is not necessarily inferior to the teacher, nor is a teacher necessarily superior to the student. —Han Yu, Shishuo (“On Teachers”) Recent progress in frontier models has renewed interest in large-scale reinforcement learning (RL) (Kimi Team, 2026; DeepSeek-AI, 2026; LLM-Core Xiaomi, 2026; Microsoft AI, 2026; OpenAI, 2026; Anthropic, 2026). Yet on testing an RL recipe on a large model requires repeated sampling and optimization, making even preliminary experimentation expensive. Rollout generation, particularly for long reasoning traces, can be a major bottleneck (Zhou et al., 2025). This cost limits how many training recipes researchers can test and makes it difficult for laboratories with modest compute budgets to explore improvements for frontier models. A compelling alternative is to develop reasoning capabilities in smaller models and transfer their benefits to stronger ones. One approach is to train specialist teachers and consolidate their capabilities into a single student through on-policy distillation (OPD), as in Kimi K3 and DeepSeek-V4 (Kimi Team, 2026; DeepSeek- AI, 2026). However, OPD still samples from and updates the student, retaining the expense of generating large-model rollouts when the student is large. Direct token-level distribution matching also assumes aligned tokenizations, although recent methods explicitly support OPD across different tokenizers (Niu et al., 2026). Our aim is to remove the need for the target strong student during 1 arXiv:2609.32913v1 [cs.LG] 26 Sep 2026 Preprint training and transfer the learned behavior through text at inference time. This leads to the central question: Can a weak teacher trained with RL improve a strong student without using any rollouts from that strong student during training? We propose Allspark, a training and inference framework for weak-to-strong transfer through alternat- ing chains of thought (Figure 1). During training, two copies of a small model take turns extending a shared reasoning trace. One copy, the weak teacher, is trained with RL; the other remains frozen and produces the final answer. At inference time, we replace the frozen training partner with a stronger student and reuse the trained teacher without further optimization. Teacher and student continue each other’s reasoning, and the student produces the final answer. The transfer interface is the shared text, allowing the same teacher to steer students with different architectures and tokenizers. The teacher therefore need not be a better standalone problem solver: its role is to contribute reasoning that helps the student reach a better answer. We study Allspark at two complementary scales. The controlled Qwen3 study uses 1.7B and 4B models (Yang et al., 2025) across math and reasoning to examine teacher-training choices and transfer to larger students under fixed inference limits. The larger-scale study trains an Inkling-Small teacher (Thinking Machines Lab, 2026b) and evaluates it on 96 development problems drawn from the public training split of ARC-AGI-2 (Chollet et al., 2025). This study examines transfer within the Inkling family and across families to Kimi K2.6 (Moonshot AI, 2026a) and Nemotron 3 Ultra (NVIDIA, 2026). It shows accuracy gains in various settings, providing evidence that a trained teacher can be reused across strong students. Using Inkling’s thinking-effort controls (Thinking Machines Lab, n.d.b), we also examine how teacher steering changes accuracy as reasoning effort varies (Figure 4). We report accuracy against retained completion tokens, making the scope of the token–performance comparison explicit. Our contributions are threefold: •We introduce Allspark, which trains a weak teacher using only small-model rollouts and reuses it to steer frozen strong students through alternating chains of thought. •We study teacher training and transfer at two scales: controlled Qwen experiments across two domains, and Inkling experiments with evidence of transfer within and across model families. • We analyze how teacher steering interacts with test-time scaling, identifying settings in which it improves accuracy and characterizing the accompanying retained-token tradeoff. 2BACKGROUND AND RELATED WORKS Weak supervision for a trainable strong student.Weak-to-strong generalization studies whether a stronger model can benefit from a weaker supervisor without inheriting its performance ceiling. Burns et al. (2023) finetune strong models on weak labels, and Yang et al. (2024) extend this approach to selected weak-model reasoning traces. These methods update the strong model, but supervised finetuning need not generate its rollouts. On-policy distillation instead gives teacher feedback on sequences sampled from the student, reducing the mismatch between training prefixes and those encountered at inference (Agarwal et al., 2024; Lu & Thinking Machines Lab, 2025). Cross-tokenizer variants can provide token-level supervision even when teacher and student vocabularies differ (Niu et al., 2026). Recent weak-to-strong distillation methods often transfer a change learned by the weak model rather than its entire policy. Direct-OPD uses the post-RL versus pre-RL log-ratio as an implicit reward on strong-student rollouts (Feng et al., 2026). W2S-OPD adds a weak-model logit contrast to the strong student’s base logits and distills the resulting proxy teacher (Yu et al., 2026). OPRD uses the weak model’s policy change to rescale the strong student’s verifier-driven policy gradient (Park et al., 2026). These approaches obtain strong-student rollouts and update its parameters during transfer. They address a different resource setting from training a weak teacher once and applying it to frozen strong students. 2 Preprint Guidance for a frozen strong student.Other methods keep the strong model fixed and alter how it is guided at inference. Weak-to-strong search steers a frozen large model with likelihood differences from tuned and untuned weak models (Zhou et al., 2024). CoWeST trains a weak collaborator using preferences derived from its interactions with a strong model; the strong model then refines the weak model’s drafts and background information (Jiao et al., 2025). Co-LLM learns when to defer to a frozen assistant (Shen et al., 2024), while weak critics can prompt a strong model to revise an answer (Jin et al., 2026). 3ALLSPARK: TRANSFER WITHOUT STRONG-MODEL TRAINING ROLLOUTS Allspark asks whether a weak model can learn to improve another model’s reasoning without using the target strong model during training. We train a weak teacher alongside a frozen weak student, initialized from the same base model. They alternate reasoning chunks, and the weak student produces the final answer. Its task reward updates only the teacher’s reasoning contributions. At inference, we freeze the teacher and replace the weak student with a frozen strong student, which now continues the shared reasoning and produces the answer (Figure 1). The teacher is thus trained entirely with weak-model rollouts and transferred through text, without updating or training on rollouts from the strong student. Shared chain of thought Answer Vanilla RLmodelmodelmodelmodelmodel Allspark small student studentteacherstudentteacherstudent Transfer strong student all frozen studentteacherstudentteacherstudent weak teacher · steers CoTupdated tokens Figure 1: Vanilla RL updates the full reasoning and answer. Allspark updates only the weak teacher’s reasoning beside a frozen small student. Transfer freezes the teacher and replaces the student with a stronger model. Blocks show authorship; underlines mark direct token loss. 3.1ALTERNATING REASONING Letπbe the weak teacher andρthe student. Given a problemx, both models extend the same reasoning prefixh, initially containingx. The student starts, and the two models take turns generating a chunk z: z ∼ ρ(·| h),student turn, π(·| h), teacher turn. (1) Each chunk is appended tohbefore the next turn. When reasoning stops, the student produces the final answer y ∼ ρ(·| h). Let R(x,y)∈0, 1 indicate whether the final answer is correct. 3.2TRAINING THE WEAK TEACHER During training,ρis a frozen copy of the teacher’s initial model. Onlyπis optimized, with the task objective max π J (π), J (π) =E π,ρ R(x,y) , ρ fixed.(2) 3 Preprint The expectation averages over training problems and the alternating rollouts above. Only sampled teacher reasoning tokens receive direct loss; student tokens and inserted boundary markers are masked out. The experiment-specific RL losses and training rewards are given in Appendices A and B.1. The usefulness of a teacher chunk can be expressed as its continuation value: Q(h,z) =E R(x,y)| h,z .(3) Here the teacher has emitted z after prefix h; the expectation covers the remaining rollout under the currentπandρ. A useful chunk therefore need not solve the problem by itself: its value comes from the student’s subsequent continuation. 3.3TRANSFER ACROSS MODEL SIZE AND FAMILY At transfer, freeze the selected teacherπand replaceρwith a stronger student. The same alternating procedure applies, with both models fixed. Transfer changes the student’s reasoning context through teacher-generated text; it does not update the student’s weights. Each writer renders and tokenizes the shared text using its own model interface, so the models need not share a vocabulary. 4EXPERIMENT We evaluate Allspark at two scales: controlled experiments with Qwen models and larger-scale experiments with Inkling models. The Qwen experiments examine weak-to-strong transfer through targeted ablations, while the Inkling experiments demonstrate improvements in the accuracy–token tradeoff and weak-to-strong transfer across model families. 4.1CONTROLLED QWEN EXPERIMENTS Setup. The controlled study uses two Qwen3-1.7B (Yang et al., 2025) training runs: ordinary RL and Allspark, each with seed 0. Both train on the same mixed-domain pool, with equal numbers of accepted prompts from each domain per update. Allspark trains alongside a frozen Qwen3-1.7B student. The resulting 1.7B teachers are reused with frozen Qwen3-1.7B and Qwen3-4B students, without additional training. Table 1 summarizes the data; Appendix A gives the loss, training recipe, and checkpoint selection. Table 1: Qwen training and development data for the two reported domains. Both methods share the screened RL pool. Development questions are disjoint from this pool but were used for checkpoint selection and subsequent evaluation. All samples come from the datasets’ original training splits. DomainDatasetRL poolDevelopment MathMATH (Hendrycks et al., 2021)142128 ReasoningLogiQA 2.0 (Liu et al., 2023)162128 Total304256 Counts are distinct prompts. Each domain starts with 512 training candidates; initial-policy screening retains prompts with 1–7 correct answers in eight samples. Each update uses two accepted prompts per domain, despite the unequal pool sizes. Controls and evaluation.Table 2 compares each frozen student alone and with the checkpoint-94 Allspark teacher, an ordinary-RL teacher, and (for the 4B student) the untrained teacher. We also include the ordinary-RL-trained 1.7B policy on its own. Every paired condition uses a Qwen3-1.7B teacher; the student remains frozen and produces the final answer. We report the math and reasoning portions of the development panel in Table 1, with 128 questions per domain and one sampled answer per question. We use final-answer accuracy for both domains. Within each comparison, we match the evaluation questions, student decoding settings, and reasoning and answer limits. 4 Preprint Table 2: Qwen development accuracy (%) on math and reasoning, with 128 questions per domain. The Allspark teacher is checkpoint 94; Ordinary RL is the separately trained Qwen3-1.7B policy. Dashes mark ongoing evaluations. StudentTeacherMathReasoning Qwen3-1.7BNone81.373.4 Ordinary RL88.271.8 Allspark89.875.0 Qwen3-1.7B (RL-tuned)None89.870.3 Qwen3-4BNone96.981.2 Untrained93.079.7 Ordinary RL89.879.6 Allspark96.182.8 Takeaway.Table 2 provides a sign of life for weak-to-strong capability transfer from an Allspark- trained teacher across math and reasoning tasks. In the 4B ablation, using either an untrained or an ordinary-RL weak teacher to steer the stronger student lowers accuracy in both domains. Using the Allspark-trained teacher instead yields approximately neutral transfer in math and positive transfer in reasoning. Case studies. Figure 2 shows two math recoveries and two reasoning answer changes on paired Qwen3-4B development questions. In the octagon problem (M3), the student alone settles on1053/2; in the alternating trace, the teacher returns to corner subtraction and the student next computes the correct area,567. In the peg-board problem (P1), the solo final never settles on a total, whereas the alternating final states the correct answer,1. Both P1 finals reach the answer-token limit, so this pair identifies an outcome difference, not a single decisive handoff. In R1 and R2, the teacher reframes the relevant claim and the student’s answer changes from incorrect to correct. The R2 teacher also overstates the passage. These selected traces show what an exchange can look like; Table 2 gives the aggregate context. Appendix C includes fuller excerpts and other cases. Figure 3 shows two complementary behaviors. In M2, the teacher supplies an algebraic substitution that the student uses in its next turn; both conditions reach the correct answer,12. In R5, the student alone selects B, while the alternating rollout selects C after the teacher rules out B’s stronger claim about ferns in thick litter. The student had considered C earlier, so this excerpt shows a useful continuation rather than an isolated moment of discovery. 4.2INKLING EXPERIMENTS Setup. We train the weak teacher, Inkling-Small (276B total parameters, 12B active per to- ken) (Thinking Machines Lab, 2026b), alongside a frozen copy of the same model. At evaluation, we replace the frozen partner with Inkling (975B total, 41B active) (Thinking Machines Lab, 2026a) as the strong student. The strong student contributes no training rollouts, and both models remain fixed during transfer. We use the 1,000 public training tasks of ARC-AGI-2 (Chollet et al., 2025), split into 904 training-pool tasks and 96 development tasks. Appendix B.1 provides the training and rollout settings. Evaluation. Using Rl’ed teacher, we compare Inkling alone, Inkling-Small with the teacher, and Inkling with the teacher (Allspark). Each configuration is evaluated at thinking efforts0.40,0.55, 0.70, and0.85, with three attempts per development task and shared task–attempt seed assignments. We report mean pass@1 over all 288 attempts and the mean retained output tokens per attempt, including both writers and the final answer. Takeaway.Allspark achieves the highest observed accuracy in Figure 4, reaching 81.6% compared with 78.1% for Inkling alone. It also improves the accuracy–token tradeoff at selected operating points: Allspark reaches 79.2% with 10.8k retained tokens, exceeding the strong student’s best 5 Preprint MATH M3 / Qwen3-4B Octagon inscribed in a square of perimeter 108 cm; find its area. Student aloneAllspark STUDENT the correct area is 526.5 cm², which is 1053/2. WEAK TEACHER the octagon is formed by cutting each corner STUDENT · NEXT 729 - 162 = 567 cm². 1053/2 Incorrect 567 Correct MATH P1 / Qwen3-4B Place 5 yellow, 4 red, 3 green, 2 blue, and 1 orange peg on a triangular board; no color may repeat in a row or column. Student aloneAllspark STUDENT FINAL However, this might not hold for all red placements. STUDENT FINAL Therefore, the final answer is 1. No settled answer Incorrect 1 Correct REASONING R1 / Qwen3-4B Genetics may cause harm. Is that the ethicist's conclusion (C), or a reason Gillette's argument is unconvincing (D)? Student aloneAllspark STUDENT Therefore, the correct answer is C. WEAK TEACHER So the conclusion is that Gillette's argument is unconvincing because he didn't consider the possible negative consequences. STUDENT · NEXT But option C is stating the reason why the argument is not persuasive. C Incorrect D Correct REASONING R2 / Qwen3-4B What role does 'learn from history' play? A: compatible; D: challenged. Student aloneAllspark STUDENT Therefore, D is not correct. The answer is A. WEAK TEACHER But the argument is against learning from history, so maybe not compatible with the conclusion. STUDENT · NEXT Therefore, D would be correct. A Incorrect D Correct Verbatim excerpts. P1 shows final answers; both P1 finals reached the token cap. Figure 2: Selected paired math and reasoning rollouts with a Qwen3-4B student and a Qwen3-1.7B weak teacher. Blue bubbles are teacher-authored; white bubbles are student-authored. M3, R1, and R2 show teacher and next-student excerpts; P1 compares final-answer excerpts. Both P1 finals hit the answer-token limit. Quotes are verbatim; task summaries are abbreviated. 6 Preprint MATH M2 / Qwen3-4B Given a+b=7 and a³+b³=91, find ab. Student aloneAllspark STUDENT 13 = (a + b)² - 3ab WEAK TEACHER a² + b² = (a + b)² - 2ab. STUDENT · NEXT (49 - 2ab) - ab = 13. 12 Correct 12 Correct REASONING R5 / Qwen3-4B Goblin ferns vanish where litter is thin and European earthworms are present. Which assumption supports a causal claim? Student aloneAllspark STUDENT Answer: B WEAK TEACHER But the argument is not about the presence of ferns in thick litters, but about the disappearance in thin litters. So B is not directly needed. STUDENT · NEXT So, after this long process, I think the correct answer is C. B Incorrect C Correct Verbatim excerpts. M2 is correct in both conditions; R5 changes B to C. Figure 3: Two further recorded handoffs with Qwen3-4B students and the same Qwen3-1.7B weak teacher. Blue bubbles are teacher-authored; other excerpts are student-authored. Both conditions solve M2; R5 changes from B (incorrect) to C (correct). Quotes are verbatim and task summaries are abbreviated. observed accuracy of 78.1% at 12.3k tokens. These results support weak-to-strong transfer without updating the strong student. 481216 Retained tokens (k) 55 65 75 85 Accuracy (%) 79.2% · 10.8k 78.1% · 12.3k Strong aloneWeak + teacherAllspark Figure 4: Accuracy–token tradeoff on 96 ARC development tasks, with three attempts per task. Paired runs reuse the rl’ed teacher with an Inkling-Small or Inkling student. Points from left to right use thinking efforts 0.40, 0.55, 0.70, and 0.85. Tokens include retained output from both writers and the final answer. 7 Preprint 4.3CROSS-FAMILY TRANSFER Setup. We reuse the same Inkling-Small weak teacher with two strong students from other model families: Nemotron-3-Ultra (550B total, 55B active) (NVIDIA, 2026), evaluated in medium- and full-thinking modes, and Kimi-K2.6 (1T total, 32B active) (Moonshot AI, 2026b) on its 128K-context serving tier. Both teacher and student remain frozen; no target-family retraining is performed. Because the models use different tokenizers and reasoning formats, each handoff decodes the retained reasoning into text and re-encodes it in the next model’s native reasoning channel. The student starts the trace and always produces the final answer. Each setting compares the student alone with Allspark on the same 96 ARC development tasks, with three attempts per task and shared task–attempt seed assignments. We report mean pass@1 and task-bootstrap intervals. The teacher uses effort0.40with Nemotron medium and0.85with Nemotron full and Kimi; their reasoning ceilings are 24,576 and 49,152 tokens, respectively. The alternating evaluator explicitly closes reasoning and requests a separate student answer, including after a reasoning-ceiling stop; the student-alone baseline must transition to its answer naturally within one completion. Thus, the comparison measures the complete inference procedure. Appendix B.2 specifies the text interface, stopping rules, and token accounting. Transfer results. We evaluate the target models at native thinking settings rather than tracing an Inkling-style effort curve. Figure 5 shows accuracy gains with different effects on retained output length. For Kimi, Allspark improves accuracy by 11.5 percentage points while using 7.7% fewer retained tokens. For Nemotron, mean accuracy increases by 17.7 and 5.2 percentage points in medium- and full-thinking modes, respectively, with increased token use. These results support reuse of the same teacher across model families, although accuracy and token benefits depends on the target student’s inference settings. 01020 +17.7 +5.2 +11.5 04080 +72.5% +4.4% -7.7% Nemotron medium Nemotron full Kimi-K2.6 Δ Accuracy (p)Δ Tokens (%) Figure 5: Changes versus each student alone: Nemotron-3-Ultra (medium/full) and Kimi-K2.6 (128K). Same weak teacher; 96 ARC development tasks, three attempts each. Bars: archived 95% task-bootstrap intervals. Tokens: retained output from both models, in native tokenizer units. 5CONCLUSION We introduced Allspark, a training and inference framework for weak-to-strong transfer through alternating chains of thought. A weak teacher learns alongside a frozen copy of the same model, then steers a stronger, frozen student at inference. This avoids strong-model rollouts during training, while the shared text interface enables transfer across model sizes and families. Our study spans math and reasoning with Qwen models and ARC-AGI-2 with larger models. The Inkling results demonstrate cross-family transfer and improved accuracy–token tradeoffs. These findings suggest a practical route to exploring RL recipes and training data with smaller models and reusing the resulting teachers with stronger students. We hope Allspark lowers the barrier to RL research and inspires extensions to model alignment and AI safety research. 8 Preprint 6LIMITATIONS Allspark avoids generating strong-model rollouts during teacher training, but inference requires serv- ing both the teacher and the student. The additional model calls and handoffs may increase inference cost and latency, so reductions in retained tokens do not necessarily translate into lower deployment cost. Our experiments provide initial evidence under limited compute resources. Evaluation covers a restricted set of models, tasks, and training runs; broader studies are needed to establish the robustness and generality of the observed gains. Finally, our implementation relies on the ability to interrupt and resume a shared reasoning stream. Its applicability to non-thinking models, or to interfaces that do not expose a suitable reasoning stream, remains untested. AI USE STATEMENT In this work, we used generative AI tools for polishing writing, drafting experiment setting since it’s mostly summarization and organization of documented experiments, and discovery of related works. We have not used generative AI tools for research ideation, generating synthetic datasets, provide core analysis in paper and proving mathematical claims. Additionally, we used generative AI tools for creating or modifying scientific figures or images and editing software code. We have reviewed all AI-assisted work. For example, we have gone through the citations found by AI and try our best to make sure they truly exist and they are truly relevant instead of hallucination. We take responsibility for the final content of this work, including text, claims or artifacts produced with the aid of generative AI. 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Yuzhen Zhou, Jiajun Li, Yusheng Su, Gowtham Ramesh, Zilin Zhu, Xiang Long, Chenyang Zhao, Jin Pan, Xiaodong Yu, Ze Wang, Kangrui Du, Jialian Wu, Ximeng Sun, Jiang Liu, Qiaolin Yu, Hao Chen, Zicheng Liu, and Emad Barsoum. APRIL: Active partial rollouts in reinforcement learning to tame long-tail generation. arXiv preprint arXiv:2509.18521, 2025. URLhttps: //arxiv.org/abs/2509.18521. Zhanhui Zhou, Zhixuan Liu, Jie Liu, Zhichen Dong, Chao Yang, and Yu Qiao. Weak-to-strong search: Align large language models via searching over small language models. arXiv preprint arXiv:2405.19262, 2024. URL https://arxiv.org/abs/2405.19262. 11 Preprint AQWEN RL IMPLEMENTATION AND TRAINING PROTOCOL A.1MODELS AND ROLLOUT MASKS Both arms initialize fromQwen/Qwen3-1.7Bwith training seed 0. Only LoRA parameters are optimized (Hu et al., 2022); the pretrained weights remain fixed. In Allspark, the student uses the same initial checkpoint without an adapter. Rollouts and update masks. Ordinary RL samples the entire reasoning and answer from the adapted model. Allspark starts with at most 128 student reasoning tokens, then alternates teacher and student chunks of at most 256 tokens until reasoning ends or reaches its shared token ceiling. Each writer conditions on the prompt and the complete preceding stream. The frozen student generates the final answer. The loss mask m t is: Token sourceOrdinary RLAllspark Sampled adapted-model reasoning11 (teacher) Sampled student reasoning—0 Sampled final answer10 (student) Forced closing token / separator00 A sampled</think>inherits its writer’s mask. A closing token inserted at the reasoning ceiling has mask zero. Student tokens remain in the causal context of subsequent teacher predictions. A single binary verifier rewardr ∈0, 1for the final answer supplies the training signal; there is no intermediate reasoning reward. A.2RL LOSS: MASKED SAMPLER-HEAD CENTERING Both arms use the same loss. Ordinary RL applies it to the adapted model’s reasoning and answer tokens; Allspark applies it only to the teacher’s reasoning tokens. The calculation has three steps. Step 1: solution advantage. For one prompt, sampleGtrajectories and grade their final answers with binary rewardsr 1 ,...,r G . Compute the group’s meanμand population standard deviationσ, then assign each trajectory its standardized reward: μ = 1 G G X j=1 r j , σ 2 = 1 G G X j=1 (r j − μ) 2 , A i = (r i − μ)/σ, σ > 0, 0,σ = 0. (4) We useG = 32trajectories per prompt. The sameA i is used at every trainable position in trajectoryi. A better-than-group-average answer has positive advantage; a worse answer has negative advantage. No epsilon is added to the standard deviation. Step 2: loss at one trainable token. Lethbe the full prefix andythe sampled next token. The current trainable model assigns probabilityp θ (v | h)to tokenv. The stored distributionq(v | h) comes from the checkpoint that sampledy, evaluated at this same prefixh. It is held fixed and may come from an older checkpoint. LetHcontain itsKhighest-probability tokens, and letAbe this trajectory’s advantage from Step 1. For this position, minimize ℓ(θ) =−A 1y ∈ H logp θ (y | h) | z sampled-token score − X v∈H q(v | h) logp θ (v | h) | z sampler-weighted center .(5) The first term scores the token that was sampled. The second is a weighted sum of the learner’s log probabilities over the saved sampler head. For an in-head sample, positive advantage favors the 12 Preprint sampled-token score relative to this center; negative advantage reverses that direction. Ify /∈ H, the first term is zero and only the centering term remains. Onlyp θ receives gradients. The advantage, sampler probabilities, and membership inHare fixed. The probabilities inHretain their original mass: they are not renormalized to sum to one. The head is used to compute the loss; generation itself does not use top-Ktruncation. We useK = 128. This is a head-only variant of score centering (Marek & Ryabinin, 2026). Unlike the published implementation, which retains the sampled-token log probability outside the head and models the tail with the learner distribution rescaled to the sampler’s tail mass, our loss sets the sampled-token term to zero when y /∈ H and omits the tail. Step 3: loss for one optimizer update. An update containsMprompt groups ofGtrajectories each. LetBbe this batch ofMGtrajectories andT (τ )the trainable positions in trajectoryτ. The update loss is L(θ) = 1 MG X τ∈B X t∈T (τ) ℓ τ,t (θ).(6) We useM = 6prompt groups (two per domain), giving 192 trajectories per update. Thus token losses are summed, and trajectories are averaged. There is no division by a trajectory’s token count. The setTimplements the mask in Appendix A.1: student and forced tokens are excluded from the Allspark loss. A trajectory with no trainable positions contributes zero loss but still enters its group’s advantage calculation and the denominator MG. A.3ASYNCHRONOUS RL AND FILTERING Three rollout actors generate trajectories while one learner updates the adapter. Each actor holds one immutable adapter snapshot for a prompt group, including screening and the fresh training rollouts. The learner pools six accepted groups (two per domain), accumulates their gradients, performs one optimizer step, and publishes a new adapter checkpoint. A batch can therefore contain groups sampled by different behavior-policy versions. Immediately before consumption, each group’s version lag must satisfy0≤ v learner − v sampler ≤ ∆ max , where∆ max is the maximum permitted lag, set to two updates. Older queued groups are discarded. Each loss-bearing token uses its own saved sampler probabilities in Equation 5. This is asynchronous off-policy optimization with bounded lag, without an importance-sampling ratio, PPO clipping, reference-policy KL term, or entropy bonus. The sampler-head term is not an importance- weight correction for stale rollouts. Prompt and group filtering. Both arms use the same frozen training pool drawn from MATH and LogiQA2. Initial-policy solo screening retains a prompt when its first complete eight-sample group has 1–7 successes. Online, each actor pins a sampling snapshot, screens a candidate with eight fresh trajectories, and requires 1–7 successes. It then draws 32 fresh trajectories using a separate random-seed namespace; groups with 1–31 successes are retained in full. Screening trajectories and tied training groups do not enter updates. Each batch contains six distinct prompts, two per domain; prompts may recur in later updates. Allspark shares student random-seed keys within a group, including the answer stage, while teacher keys vary across trajectories. A.4SOLUTION-LENGTH CONTROLS AND TERMINATION Length is controlled by separate hard generation ceilings for reasoning and the final answer. The reasoning ceiling is shared across all chunks from both writers, not allocated afresh to each writer or turn. Each call is limited by the remaining budget; Allspark additionally limits its first student chunk to 128 tokens and later chunks to 256. All returned raw tokens are retained in the shared stream, without paragraph trimming or prefix cropping. 13 Preprint Limit (tokens)Training / historical evaluationRevised evaluation Total context8,19216,384 Shared reasoning4,0968,192 Final answer1,5364,096 A sampled</think>ends reasoning naturally. If the reasoning ceiling is reached first, the generator inserts</think>with zero loss mask and starts the answer phase. An EOS or empty generation during reasoning instead invalidates the answer stage. The answer stops at its own ceiling or native EOS; unused reasoning budget is not transferred to the answer allowance. Before generation, the full prompt, both allowances, and required boundary tokens must fit the context window; an oversized request is rejected rather than silently cropped. There is no minimum generation length, length-dependent reward penalty, or per-sequence length normalization in the loss. Equation 6 sums trainable-token terms, so different trajectory lengths contribute different numbers of terms. A cap hit is recorded, but does not itself force zero reward: the saved final answer is scored by the verifier. Report actual reasoning, answer, and trainable-token counts and both cap rates; identical ceilings do not imply identical realized length or compute. The longer evaluation protocol changes inference only and cannot repair truncated training rewards. A.5OPTIMIZATION SETTINGS Gradients accumulate one trajectory at a time over the accepted batch, followed by one clipped optimizer step. Log probabilities and loss arithmetic use FP32; the base model is loaded in BF16. Causal forward passes retain all prefixes preceding loss-bearing tokens, with gradient checkpointing. Vocabulary projections are evaluated in chunks of 32 loss-bearing positions. Table 3: Shared settings for the local Qwen training runs. SettingValue OptimizerAdamW; constant learning rate 10 −5 Adam moments / epsilon(0.9, 0.999) / 10 −8 Weight decay / gradient-norm ceiling 0 / 1 LoRA rank r / α / dropout16 / 32 / 0 LoRA projectionsq,k,v,o, gate, up, down; no biases Training context ceiling8,192 tokens Reasoning / answer ceilings4,096 / 1,536 generated tokens SamplingTemperature 1; top-p = 1; top-k disabled Sampler head used in the loss128 tokens; no generation truncation Queue capacity12 prompt groups, four per domain A.6DEVELOPMENT EVALUATION AND CHECKPOINT SELECTION The frozen development panel contains 384 questions, with 128 each from the MATH, TACO, and LogiQA 2.0 training splits, disjoint from the local training pool. Each candidate is evaluated as a teacher alternating with the same frozen, unadapted Qwen3-1.7B student; the student writes the final answer. There is one sampled answer per question (seed 0), using the revised evaluation limits in Appendix A.4 and the same sampling settings and chunk schedule. The training limits remain unchanged. Checkpoint selection rule. All 384 development questions are evaluated for each candidate. Let N d be the number of development questions in domaindandc d (s)the number correct at checkpoint s. Selection is s ∗ = min argmax s∈C c math (s) + c reasoning (s) N math + N reasoning .(7) HereN math = N reasoning = 128. Exact ties choose the earlier update. Reasoning accuracy and the three-domain mean are reported separately and do not choose the checkpoint. Ordinary-RL update 14 Preprint 92 is the nominated comparison checkpoint. . The completed review selects checkpoint 94 (Table 4); Table 4: Finalized Allspark checkpoints under the revised evaluation protocol. Entries are correct answers on the two reported domains; each has 128 development questions. Checkpoint 94 is the fixed selection from the original three-domain protocol. Bold marks the selected checkpoint. These are development results, not public-test scores. CheckpointMathReasoning 5511693 9111496 9411996 it is held fixed for subsequent evaluations and transfer to the 4B student. Ordinary-RL checkpoint 92 retains its earlier nomination. A.7DEVELOPMENT LEARNING CURVES 0102050100 Step (log scale) 80 85 90 Dev accuracy (%) Math 0102050100 Step (log scale) 65 70 75 80 Reasoning Ordinary RL Figure 6: Development accuracy of the ordinary-RL-trained Qwen3-1.7B teacher alternating with the frozen student, under the historical 8K protocol (4,096 reasoning and 1,536 answer tokens). Each domain has 128 fixed questions; the figure reports math and reasoning only. Filled circles connect the initial checkpoint (step 0) to checkpoints 10, 20, 50, and 100 without smoothing. 15 Preprint A.8TRAINING DIAGNOSTICS 110100 Step (log scale) 20 40 60 80 Training pass (%) Math 110100 Step (log scale) 20 40 60 80 Code 110100 Step (log scale) 20 40 60 80 Reasoning 110100 Step (log scale) 30 40 50 60 Training pass (%) Overall 110100 Step (log scale) 0.30 0.33 0.36 Entropy (nats) Learner entropy 5-update trailing means Ordinary RL Allspark Ordinary RL Allspark Figure 7: Qwen3-1.7B training diagnostics over the first 100 committed updates of each method on a logarithmic step axis (one seed per method). Lines show full trailing-five-update means, starting at update 5. Pass rates are measured on accepted, filtered training rollouts; overall is the equal-domain mean. Learner entropy is measured on the respective trainable-token contexts. The accepted prompts and rollout policies differ between methods, so these curves are descriptive training diagnostics, not held-out accuracy or paired treatment effects. Allspark completed 100 updates; the ordinary-RL run stopped at 110. Figure 7 shows the common 1–100 range. The additional ten ordinary-RL updates remain part of its training expenditure. Update counts do not equate generated tokens, optimized tokens, or GPU time; screening and rejected rollouts also contribute to training cost. 16 Preprint BINKLING TRAINING PROTOCOL B.1INKLING TRAINING AND INFERENCE Training. We trained a rank-32LoRA adapter (Hu et al., 2022) on Inkling-Small for72rollout steps (indexed0–71), paired with a frozen copy of the same base model. The frozen copy opened each reasoning trace and generated the visible answer. Each step sampled16tasks at all three thinking efforts,0.4, 0.55, 0.7, with32rollouts per task–effort group: nominally1,536training rollouts. We used four minibatches, keeping each group together, with one optimizer update per nonempty minibatch. The learning rate was 2× 10 −5 through step 24 and 5× 10 −5 thereafter. RL loss. Because training used the Tinker API, we chose its built-inimportance_sampling policy-gradient objective (Thinking Machines Lab, n.d.a). Each trajectory contributes ℓ(π) =− ̄n n A X t∈T π(y t | h t ) π old (y t | h t ) , A = r− ̄r.(8) Hereπis the current teacher,π old is the teacher that sampled the rollout, andh t is the full shared prefix before tokeny t .Tcontains only teacher-generated reasoning positions,n =|T|, and ̄nis the mean teacher-token count across retained trajectories in the rollout batch. The factor ̄n/nnormalizes each trajectory’s contribution by its trainable length. The baseline ̄ris the mean reward within the same task–effort group; we do not divide by the group standard deviation. We sum these losses within each minibatch. The prompt, frozen-model chunks, and final answer receive no policy-gradient loss. Reward and length control.Letsindicate exact task success,cdenote cell accuracy, andLcount retained reasoning tokens from both writers, excluding the final answer. We use r = ws + (1− w)c− min p max ,λ max 0, L L 0 − 1 .(9) We setw = 0.85,L 0 = 5,000, andp max = 0.30. Steps0–35useλ = 0; steps36–71continue the same adapter withλ = 0.15. Thus the length penalty is zero through5,000reasoning tokens, then increases by0.15per additional5,000tokens, up to0.30. Training also imposes a hard reasoning ceiling of24,576tokens; on reaching it, the reasoning channel is closed and the frozen model generates the final answer. Models and rollout settings. All Inkling models are served through Tinker. The teacher and its frozen training partner usethinkingmachines/Inkling-Small; the strong transfer student isthinkingmachines/Inkling. A deterministic task-ID hash defines the 904/96 training- pool/development split. The student opens each trace with at most 256 tokens, followed by alternating teacher and student chunks of at most 2,048 tokens. Handoffs are trimmed to the last paragraph boundary when at least half the sampled chunk can be retained. The frozen student produces the final answer as a JSON object containing the predicted output grids. Both writers use temperature1.0 and top-p 1.0, with a final-answer allowance of 4,096 tokens. In the checkpoint-63 effort sweep, the reasoning ceiling is 24,576 tokens at efforts 0.40, 0.55, and 0.70, and 49,152 tokens at effort 0.85. B.2CROSS-FAMILY INFERENCE PROTOCOL Models and operating points. The cross-family evaluation pairs the Inkling-Small teacher from the within-family study with frozen Nemotron-3-Ultra or Kimi-K2.6 students. Neither the teacher nor the target student is updated during these evaluations. Table 5 lists the three operating points; the two Nemotron rows are settings of one model family. The numerical effort parameter applies only to Inkling-Small. The target students instead use their native thinking renderers, so equal numerical effort or equal reasoning compute across families is not assumed. Text handoffs. Each model receives the same ARC task messages, rendered in its own format. Every reasoning chunk is decoded with its writer’s tokenizer, stripped of model-specific control markers, and appended to a shared text trace. Before the next turn, the complete trace is re-encoded with the receiving model’s tokenizer and placed in its native hidden-reasoning channel. This preserves 17 Preprint Table 5: Cross-family inference settings. Context and reasoning limits are in tokens. The context limit applies to the student; the reasoning guard is checked in the receiving model’s tokenizer at each handoff. Student settingTeacher effortStudent contextReasoning limit Nemotron-3-Ultra, medium0.4065,53624,576 Nemotron-3-Ultra, full0.8565,53649,152 Kimi-K2.6, 128K0.85131,07249,152 the retained reasoning text while flattening model-specific message boundaries; token IDs are never passed directly between families. The student starts the trace, subsequent turns alternate between teacher and student, and the student always generates the final answer. The evaluation configuration specifies a 256-token first chunk, 2,048-token later chunks, paragraph trimming that retains at least half of a sampled chunk, temperature 1.0, top-p 1.0, and a final-answer allowance of 4,096 tokens. Stopping and grading.At each handoff, the evaluator recomputes the trace length in the receiving model’s tokenizer and limits the next chunk by both the reasoning ceiling and the remaining context, reserving space for the answer. Either model can end reasoning with its native closing marker. Alternation also stops when the reasoning allowance is exhausted. The evaluator then closes the hidden channel in the student’s format and requests a separate final answer, including after a reasoning- ceiling stop. By contrast, the student-alone baseline generates one uninterrupted completion with a total allowance equal to its reasoning room plus the answer allowance; it must emit its own hidden-to-visible boundary. Only the visible final answer is graded against the exact ARC output grids. Aggregation and scope. Each condition contains 288 attempts over the same 96 development tasks. Mean pass@1 averages exact success over attempts; it is not best-of-three accuracy. For student-alone comparisons, task–attempt seed assignments are paired. The archived 95% intervals use 5,000 paired bootstrap resamples of task-level mean differences, with bootstrap seed 17. The Inkling-Small–teacher reference is reused from the within-family scan; its medium-effort comparison is task matched but not seed matched. Token totals sum retained teacher and student output, including the final answer, in each writer’s native tokenizer units; discarded chunk suffixes are excluded. These totals are neither a common-tokenizer budget nor total billed generation. A separate 16-task target-model pilot examined thinking modes and context limits, including an earlier 32K Kimi configuration. B.3EARLIER WITHIN-FAMILY CONTROL This evaluation uses checkpoint 35 at effort0.7, with one attempt on each of the 96 development tasks. Each frozen student (Inkling-Small or Inkling) is evaluated alone, with an untrained Inkling-Small teacher, and with the checkpoint-35 Allspark teacher. All model weights are fixed during evaluation. Table 6: Step 35, effort 0.7, one attempt on each of 96 ARC development tasks. StudentTeacherAccuracy (%) Inkling-SmallNone60.4 Inkling-SmallUntrained60.4 Inkling-SmallAllspark-trained70.8 InklingNone75.0 InklingUntrained66.7 InklingAllspark-trained76.0 B.4DOLLAR COST AND LATENCY Cost model. We report the inference-cost estimates preserved in the Tinker experiment archive. They use the rates in Table 7, which apply the archive’s recorded 50% account discount to its list-price 18 Preprint snapshot. These are historical analysis assumptions, not current price quotes. The calculation assumes a separate prefix cache for each writer: each model prefills the prompt on its first turn, pays the cached-read rate for its existing prefix on later turns, and pays fresh-prefill rates for newly received text. Retained output, including the final answer, is charged at the generating model’s sampling rate. Table 7: Historical rate assumptions, in USD per million tokens. Model / serving tierFresh prefillCached readSampling Inkling-Small0.5800.0581.440 Inkling1.8700.1874.680 Nemotron-3-Ultra, 64K2.4900.2496.225 Kimi-K2.6, 128K2.5750.5156.405 Table 8: Archived modeled inference cost, in USD per attempt, over 96 development tasks with three attempts each. Weak + teacher always uses an Inkling-Small student and the trained teacher; the cross-family rows reuse that reference. For the Inkling rows, effort applies to both writers; for cross-family rows it denotes the teacher’s effort, with the target student using its native thinking setting. Student settingEffortStudent aloneWeak + teacherAllspark Inkling0.400.02660.01980.0483 0.550.03110.01990.0546 0.700.04260.02320.0606 0.850.06230.03540.0895 Nemotron, medium0.400.04590.01960.0757 Nemotron, full0.850.12300.03520.1407 Kimi, 128K0.850.14660.03520.1774 At the same Inkling effort, Allspark’s modeled cost per attempt is1.42–1.82×that of the student alone. Across different efforts, the highlighted Inkling comparison uses 12.1% fewer retained tokens (10,771 versus 12,253), but only about 2.7% less modeled cost ($0.0606 versus $0.0623). The cross- family estimates are higher for Allspark in all three settings. In particular, Kimi’s 7.7% reduction in retained tokens accompanies a 21.1% increase in modeled cost, from approximately $0.1466 to $0.1774 per attempt. Prefix processing, cached rereads, and the mix of teacher and student tokens therefore affect the dollar comparison. 19 Preprint CROLLOUT CASE STUDIES The cases below are selected paired development rollouts. They illustrate the content exchanged between a frozen student and the weak teacher; they do not establish the causal effect of an individual teacher span. We include successful, unchanged, ambiguous, and unsuccessful pairs. Selection context. The cases include the two domains reported in Table 2 and two supplementary coding rollouts. They cover unchanged, recovered, and unsuccessful outcomes. Full task identities, excerpt role offsets, and source hashes are retained in the accompanying case manifest; complete traces remain in the run archive. CaseDomainStudentNative→ Allspark M1Math4Bcorrect→ correct M2Math4Bcorrect→ correct M3Math4Bincorrect→ correct R1Reasoning4Bincorrect→ correct R2Reasoning4Bincorrect→ correct C1Coding4Bincorrect→ correct C4Coding4Bcorrect→ incorrect 20 Preprint M1 | Math | Qwen3-4B The unpaired square root TASK SUMMARY Name the greatest whole number less than 100 that has an odd number of positive factors. Student aloneAllspark · alternating reasoning STUDENT · EXCERPT Oh! I think it's when the number is a perfect square. Because in a perfect square, one of the factors is repeated. STUDENT · TURN 1 I remember that numbers usually have factors in pairs, right? Like for a number n, if d is a factor, then n/d is also a factor. WEAK TEACHER · TURN 2 If a number is a perfect square, then one of its factors is the square root, and that factor pairs with itself. STUDENT · TURN 3 Wait, according to the logic, if all factors come in pairs, then the number of factors is even unless there is a square root factor. Therefore, only perfect squares have an odd number of factors. 81 Correct 81 Correct READING THE EXCHANGE The teacher names the self-pairing square root and the student carries the criterion forward to 81. math:EleutherAI/hendrycks_math:number_theory:261 Exact saved excerpts · teacher 1.7B / step 94 · selected development pair Figure 8: Case M1: The unpaired square root. Recorded excerpts; task text is summarized above. 21 Preprint M2 | Math | Qwen3-4B Eliminate the square terms TASK SUMMARY Two reals a and b are such that a+b=7 and a³+b³=91. Compute ab. Student aloneAllspark · alternating reasoning STUDENT · EXCERPT 13 = (a + b)² - 3ab STUDENT · TURN 1 Oh yeah, the formula is a³ + b³ = (a + b)(a² - ab + b²). Right? So maybe I can use that here. WEAK TEACHER · TURN 2 a² + b² = (a + b)² - 2ab. Since a + b is 7, that would be 7² - 2ab, which is 49 - 2ab. STUDENT · TURN 3 (49 - 2ab) - ab = 13. So combining like terms: 49 - 3ab = 13. 12 Correct 12 Correct READING THE EXCHANGE The teacher substitutes for the squared terms, and the student combines them to obtain ab=12. math:EleutherAI/hendrycks_math:intermediate_algebra:42 Exact saved excerpts · teacher 1.7B / step 94 · selected development pair Figure 9: Case M2: Eliminate the square terms. Recorded excerpts; task text is summarized above. 22 Preprint M3 | Math | Qwen3-4B Recovery after uncertain geometry TASK SUMMARY An octagon is inscribed in a square so that the vertices of the octagon trisect the sides of the square. The perimeter of the square is 108 centimeters. What is the number of square centimeters in the area of the octagon? Student aloneAllspark · alternating reasoning STUDENT · EXCERPT Given that the shoelace formula is reliable and I've double-checked it, I think the correct area is 526.5 cm², which is 1053/2. STUDENT · TURN 23 Wait, maybe each triangle has legs of 9 cm? Let me think. WEAK TEACHER · TURN 24 Alternatively, maybe the octagon is formed by cutting each corner STUDENT · TURN 25 The area of this triangle is (9 * 9)/2 = 40.5 cm². Similarly, each corner has such a triangle. There are four corners, so total area cut off is 4 * 40.5 = 162 cm². Then, the area of the square is 27*27 = 729 cm². Therefore, the area of the octagon would be 729 - 162 = 567 cm². But wait, is that correct? 1053/2Incorrect567Correct READING THE EXCHANGE The teacher returns to corner subtraction and the student computes 567, which matches the eventual final answer. math:EleutherAI/hendrycks_math:geometry:89 Exact saved excerpts · teacher 1.7B / step 94 · selected development pair Figure 10: Case M3: Recovery after uncertain geometry. Recorded excerpts; task text is summarized above. 23 Preprint R1 | Reasoning | Qwen3-4B Separate a reason from the conclusion TASK SUMMARY An ethicist rejects Gillette's promise of genetic research benefits because it ignores possible harms. What is the conclusion? C: research might harm more than help. D: Gillette's argument is unconvincing because it ignores possible consequences. Student aloneAllspark · alternating reasoning STUDENT · EXCERPT Therefore, the correct answer is C. Because the ethicist is saying that the harm is a possibility, which is the main point. The answer is C. STUDENT · TURN 1 So the question is asking for the conclusion of the ethicist's argument. The conclusion is the main point the ethicist is trying to make. WEAK TEACHER · TURN 2 The ethicist is saying that Gillette's argument isn't good because he ignores possible harms. So the conclusion is that Gillette's argument is unconvincing because he didn't consider the possible negative consequences. STUDENT · TURN 3 But option C is stating the reason why the argument is not persuasive. However, the conclusion of the ethicist's argument is that Gillette's argument is not persuasive. CIncorrectDCorrect READING THE EXCHANGE The teacher identifies the criticism as the conclusion, and the next student turn distinguishes it from the supporting reason. reasoning:csitfun/LogiQA2.0:13825 Exact saved excerpts · teacher 1.7B / step 94 · selected development pair Figure 11: Case R1: Separate a reason from the conclusion. Recorded excerpts; task text is summarized above. 24 Preprint R2 | Reasoning | Qwen3-4B Find the claim being challenged TASK SUMMARY The passage questions whether history's lessons can be discovered or applied. What role does 'we should learn the lessons of history' play? A: compatible with accepting or denying the conclusion. D: the position the argument aims to discredit. Student aloneAllspark · alternating reasoning STUDENT · EXCERPT Therefore, D is not correct. The answer is A. Because the position is compatible with the conclusion. STUDENT · TURN 1 The question is about how the idea that we should learn from history is used in the argument. The options are A to D. Let me break down each part. WEAK TEACHER · TURN 2 But the argument is against learning from history, so maybe not compatible with the conclusion. Wait, the conclusion is that we shouldn't learn from history, so if the argument's conclusion is that we shouldn't learn, then the idea of learning is being challenged. So maybe A is not correct. STUDENT · TURN 3 So the author is arguing against the idea that learning from history is useful. So the argument is directed towards discrediting the idea that we should learn from history. Therefore, D would be correct. AIncorrectDCorrect READING THE EXCHANGE The teacher frames the historical-lessons claim as the target of criticism, and the next student turn selects D. reasoning:csitfun/LogiQA2.0:10768:row:train:855c7899398a7d2c3d14b2e6c26679fc2def1e70703ffa71ca3fb9f3df406ca Exact saved excerpts · teacher 1.7B / step 94 · selected development pair Figure 12: Case R2: Find the claim being challenged. Recorded excerpts; task text is summarized above. 25 Preprint C1 | Coding | Qwen3-4B Parity before an exception TASK SUMMARY Remove as many tree edges as possible while leaving every connected component even-sized; return -1 if impossible. Student aloneAllspark · alternating reasoning STUDENT · EXCERPT if n == 1: print(0) return STUDENT · TURN 1 The key is to find how many such splits are possible. Let's think about the conditions. WEAK TEACHER · TURN 2 An important observation is that for the entire tree to be split into components with even sizes, the total number of vertices must be even. STUDENT · TURN 3 Wait, but the original tree has n vertices. So, if n is odd, then it's impossible. Because even if you split into multiple components, each of even size, the sum would be even. FINAL CODE · EXCERPT if n % 2 != 0: print(-1) return 0 for n=1 (incorrect)Incorrect-1 for n=1; 96/96 supplied tests Correct READING THE EXCHANGE The teacher introduces the even-total invariant, the student adopts it, and the final code rejects the one-node case. code:BAAI/TACO:train:16753 Exact saved excerpts · teacher 1.7B / step 94 · selected development pair Figure 13: Case C1: Parity before an exception. Recorded excerpts; task text is summarized above. 26 Preprint C4 | Coding | Qwen3-4B A shared indexing error TASK SUMMARY Convert an integer from 0 through 99 to its lowercase English spelling. Student aloneAllspark · alternating reasoning STUDENT · EXCERPT print(f"tens_words[tens_digit]-o nes_words[ones_digit]") STUDENT · TURN 3 So the code needs to have two dictionaries or lists: one for the ones (0-19), and one for the tens (2-9). WEAK TEACHER · TURN 4 tens = ["", "twenty", "thirty", "forty", "fifty", "sixty", "seventy", "eighty", "ninety"] STUDENT · TURN 5 For example, 20: s//10 is 2, so index 0 in the tens list. ninety-nine for 99; 200/200 supplied tests Correct eighty-nine for 99 (incorrect) Incorrect READING THE EXCHANGE The teacher supplies an inconsistent tens table and the student repeats the indexing mistake in the final code. code:BAAI/TACO:train:17287 Exact saved excerpts · teacher 1.7B / step 94 · selected development pair Figure 14: Case C4: A shared indexing error. Recorded excerpts; task text is summarized above. 27