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Abstract: Sampling and selection spend test-time compute inside a candidate pool, but they cannot recover a correct answer that never enters it. We introduce ValueFormation, a test-time policy that decides after each primary-model draw whether to stop, continue sampling, or acquire one proposal from a heterogeneous reasoning specialist. The three actions are compared on one cost-regularized value scale learned from vote-state features; the calibration is frozen across tasks and primary models, and acquired proposals are accepted only through a frozen posterior comparison. On held-out MATH-500, one Llama-fitted calibration improves over the strongest of eight identically evaluated baselines by more than fifteen points for each of three 8B primaries and transfers unchanged to Qwen3-14B. Removing acquisition lowers Llama MATH accuracy by 20.43 points, while same-trace and same-access replays attribute the gain to state-conditioned acquisition rather than specialist access alone. Where primary coverage is high, as on GSM8K, acquisition is rare and gains mainly come from cheaper allocation; on MBPP, a fixed code route attains the top accuracy at the lowest interaction cost. Varying the token price monotonically reduces calls without retraining.

Abstract: Inference-time scaling enhances a model's reasoning by extending its chain-of-thought (CoT). However, existing approaches typically rely on a single policy trained with outcome-reward reinforcement learning (RL), which often suffers from long-horizon plan failures where the implicit plan drifts from valid strategies, especially for small LMs with limited capacity. To address this, we propose Multi-Level Reasoning (MLR), which reformulates long-CoT generation as a two-level stochastic process. A high-level planner generates structured step descriptors specifying both the reasoning mode and the semantic subgoal. The low-level executor then produces detailed reasoning conditioned on these descriptors, forming an alternating plan--execute loop. To maintain scalability, we adopt a minimal design where the base model serves as the low-level policy and a lightweight LoRA module implements the high-level policy. For training, we observe that outcome-reward RL provides sparse and delayed feedback for long trajectories (e.g., several thousand tokens), hindering credit assignment. We therefore introduce iterative Step-DPO, a process-level preference optimization scheme that leverages Twisted Sequential Monte Carlo (TSMC) to provide scalable stepwise supervision. This yields more effective training, improved stability, and higher accuracy. Extensive experiments on challenging math, science, and logical reasoning benchmarks show that, under the same reduced data budget (10% SFT and 5% preference relative to the DeepSeek-R1 distillation setup), MLR outperforms both SFT-based distillation and strong RL/preference-optimization baselines across multiple base models and tasks. Moreover, MLR exhibits slower performance degradation on long-horizon reasoning, demonstrating stronger robustness under extended CoT generation.

Abstract: Continuous latent reasoning enables language models to perform intermediate computation without generating explicit reasoning traces. Advances in stochastic latent reasoning allow multiple paths to be sampled in continuous space, but stochasticity alone does not ensure effective exploration. A key challenge is guiding stochastic transitions between intermediate latent states toward useful reasoning paths. We introduce SubLAT, which couples a relative answer potential with Subtrajectory Balance to learn stochastic latent exploration. The relative potential evaluates the answer support of sampled states against replacement states, with a correctness signal at the terminal state. Subtrajectory Balance relates potential differences to forward and backward transition densities across subtrajectories of different lengths. The model is jointly trained with answer supervision, CoT supervision, and representation distillation. Experiments on arithmetic and deductive reasoning show increasing solution coverage with larger sampling budgets across in-domain and out-of-domain benchmarks, as well as improvements on several structural OOD benchmarks. Trajectory analyses reveal progressive divergence among sampled latent paths and higher mean relative potentials for correct trajectories across arithmetic benchmarks.

Abstract: Can a model learn by trying to fool another model? We show that it can. Inspired by the Turing Test and its recent generalization to arbitrary distinguishers, we introduce a domain-based teacher-Turing-Test, in which a stronger teacher model serves as both the target of imitation and the distinguisher, and a weaker student model is trained to make the teacher attribute responses to itself. We use this self-attribution (or fooling) probability as a reinforcement learning reward, without ever providing the student ground-truth answers, nor requiring the teacher to generate responses. We test our approach independently in four domains with verifiable answers: CommonsenseQA (reasoning), GSM8K (mathematics), KodCode (programming), and Spider (SQL), training a Qwen2.5-1.5B student/imitator model using DeepSeek V4 Flash as teacher/distinguisher. Beyond improving at passing the Turing Test, the student improves on independent, held-out evaluations in all domains and runs. To probe why being mistaken for the teacher can improve capability, our analyses reveal that self-attribution is associated with correctness -- substantially more than formatting changes -- and that it rewards partial correctness. Our results demonstrate a new, simple teacher-student training paradigm with a private and/or high-cost teacher. They also suggest that indistinguishability need not merely measure behavior -- it can also be a mechanism for learning.

Abstract: Mathematical problem solving and autonomous agents are increasingly important applications of language models, demanding stronger reasoning capabilities. Computational depth is a key resource for such reasoning, but adding model layers increases both training and inference costs, while test-time scaling incurs substantial additional inference computation. We introduce P2N (Previous to Next), which reuses the previous token's deep representations, already available during generation, to increase effective computational depth with negligible decoding overhead. Specifically, P2N adds the previous token's deep hidden state to the current token's shallow representation, allowing subsequent layers to build on previously completed deep computation. This recurrence extends effective computational depth across tokens, while introducing only a single vector addition per decoding step. We train the recurrence using parallel Jacobi updates restricted to a middle-layer core. Experiments with Qwen-style models show that, at 1.5B parameters, P2N improves GSM8K accuracy by over 10 percentage points under both zero-shot and few-shot prompting, with nearly unchanged decoding latency. The benefits also extend to language modeling and average performance on general downstream benchmarks. on general downstream benchmarks.

Abstract: Sampling multiple chain-of-thought rationales from a language model can uncover correct solutions missed by a single attempt. However, these rationales are typically lengthy, making repeated autoregressive generation costly. Continuous latent reasoning offers a compact alternative, but point-valued methods condition repeated answer sampling on a single rationale, which can limit exploration of diverse paths and answer coverage. We thus propose rationale distribution distillation (RDD), training a conditional flow over fixed-length latent sequences by distilling from multiple textual reasoning paths. Furthermore, we expand the distillation targets through within-query mixup, interpolating between complete latent rationales for the same query. The flow nonautoregressively generates the entire latent sequence, and a frozen language model autoregressively decodes answers from sampled rationale contexts. Empirically, the resulting distributional reasoner achieves higher answer coverage than the evaluated point-valued latent reasoning baselines, particularly on advanced mathematical tasks.

Abstract: On-policy distillation (OPD) provides teacher supervision on student-generated trajectories, but a single trajectory realizes only one continuation at each decision, limiting supervision coverage under a finite sampling budget. We propose BranchOPD (Branching On-Policy Distillation), which selects informative decision points using student uncertainty and teacher-student disagreement, resamples student-policy continuations from shared prefixes, and applies token-level teacher supervision to both main trajectories and branches. BranchOPD operates at token-level decisions in mathematics and within-turn decisions in agent tasks. Shared-prefix reuse reduces student-generated token costs by approximately 19% on ALFWorld and 31% on WebShop relative to seven independent trajectories. We also introduce protocol-boundary repair (PBR), which uses targeted supervision under protocol constraints to mitigate repetition and wasted generation. BranchOPD outperforms Vanilla OPD and Multi-Rollout OPD (n=7) across all evaluated benchmarks, with gains over Vanilla OPD of 18.55 percentage points in mean mathematics accuracy over 16 sampled answers (avg@16) and 13.8 points in mean ALFWorld ID/OOD success. Mechanistic analyses and selection ablations show that branching at uncertain decisions expands subsequent behavior coverage and outperforms random selection and turn-initial branching.

Abstract: Sampling several responses and voting over their answers can improve a language model's accuracy, but repeated answers limit the benefit of additional samples. Raising temperature increases diversity at a potential cost to per-sample accuracy. We introduce Self-Repulsion (SR), a sampler for masked diffusion language models that uses peer commitments to diversify the pool. At each penalized denoising step, each path lowers a token's logit according to how many peers have committed that token at the same position. Paths share a batched forward pass and then commit in sequence, so later paths observe choices made earlier in the same step. This coupling requires no training or additional forward or backward pass and can produce distinct paths even at temperature zero. When all paths commit a position together from identical logits, the update exactly maximizes total logit minus a convex duplication cost. On LLaDA-8B-Instruct with ten paths and 128 denoising steps, deterministic SR reaches 80.38% plurality accuracy on GSM8K, compared with 70.17% for the unpenalized greedy decoder. At temperature 0.6 and matched model-evaluation budgets, the count penalty improves over self-consistency by 2.06 percentage points in blocks of 32 and 14.50 under pure diffusion. Experiments on GSM8K, MATH and TruthfulQA show that voting gains arise mainly from higher coverage of correct answers, with gains that vary by benchmark and decoding regime.

Abstract: On-policy self-distillation trains reasoning models on their own trajectories using dense token distribution guidance from a privileged teacher with access to a verified solution. This supervision transfers what the teacher predicts without directly transferring where it attends within the preceding context. We introduce On-Policy Attention Self-Distillation (OPASD), which complements token-level supervision with solution-conditioned attention distillation. Because the privileged teacher can attend to verified solution tokens unavailable to the student, OPASD projects teacher attention onto student-visible positions and renormalizes the resulting distribution before alignment. Across three model sizes and four competition-level mathematics benchmarks, OPASD consistently outperforms token-only OPSD, improving average accuracy by 4.98 to 8.40 percentage points. OPASD also avoids the response-length inflation and performance degradation observed with token-only distillation, reducing generated rollout tokens by 73.9% and estimated model compute by 72.6% while training 1.53x faster. These results show that solution-conditioned attention provides a complementary supervision signal that makes on-policy self-distillation more accurate, stable, and compute-efficient.

Abstract: This empirical study is an independent reproduction of the dissociation Zhao reported in 2026. The shape of a large language model's chain-of-thought entropy trajectory predicts whether the final answer is correct, while the magnitude of its total entropy drop does not. The dissociation merits reproduction because the magnitude half rests on a single 300-problem run with one model at one seed, while the shape half was reported at full scale on both benchmarks and on a second model family. Registered at OSF before any confirmatory run, the reproduction crosses the complete GSM8K and MATH-500 benchmark test sets with four open-weight models including one reasoning-distilled model of a kind the original did not test. The shape signal replicates. The magnitude signal divides by setting. On the anchor model the accuracy gap between monotone and non-monotone chains is +9.6 percentage points on GSM8K and +27.5 on MATH-500, while the rank correlation of the total entropy drop with correctness is -0.018 on GSM8K and +0.414 on MATH-500. On the reasoning-distilled model the binary form of the shape signal fires on about one chain in a hundred, too few to estimate the registered contrast, while the graded violation count remains predictive there. In an exploratory comparison the final-step entropy alone outperforms the binary shape flag in all eight model-by-benchmark cells by ROC area, and in six or seven by the risk-coverage area the original reports, depending on an integration range the original does not state. The study contributes a reproduction of the shape signal at full test-set scale under seven documented protocol differences, a map of the settings where the magnitude signal holds and fails, and measurements of four protocol dependencies the original does not report.

Abstract: Large language models solve grade-school math word problems with high accuracy, yet a single irrelevant clause inserted into the problem can collapse it. We reconcile these observations with a mechanistic account. We show that the model's internal computation decomposes into a four-stage sequential pipeline, Schema Abstraction, Operation Planning, Operand Binding, and Computation, each stage producing a distinct intermediate representation in an identifiable band of layers. Using the same scaffold to diagnose distractor-induced failure, we localize the corruption to a single stage, Operation Planning, implemented by a set of attention heads whose causal role we validate bidirectionally. In short, we provide a mechanistic interpretation of math word problem reasoning in LLMs, and their failure when distracted.

Abstract: The reasoning ability of large language models (LLMs) is a critical factor for practical LLM-based applications. To investigate the current reasoning capability of LLMs, we clarify the types of errors that arise in LLMs' reasoning processes on mathematical datasets. We focus on problems where LLMs produce an incorrect answer. We define errors in the reasoning process as reasoning errors and manually analyze the features of reasoning errors. We defined and classified 21 error classes and identified the frequently occurring classes among them. Beyond qualitative evaluation, we leverage the evaluation results to improve the reasoning capability. We designed a prompt that explicitly focuses on eight error classes. The experiments demonstrate that this prompt effectively improves reasoning performance. Furthermore, the results suggest that the frequent reasoning errors identified in this paper are common across LLMs of comparable scale.