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ViRC: Enhancing Visual Interleaved Mathematical CoT with Reason Chunking

Lihong Wang, Liangqi Li, Weiwei Feng, Jiamin Wu, Changtao Miao, Tieru Wu, Rui Ma, Bo Zhang, Zhe Li

arXiv:2512.14654v3cs.CV

TL;DR

Existing MLLMs often reason from a single static mathematical image, while visual CoT can add redundant visual signals and lacks explicit hierarchical decomposition. ViRC structures reasoning into CRUs, trains with CRUX and a progressive strategy, and reports an 18.8% average gain over baselines on multimodal mathematical benchmarks.

  • Problem

    Existing MLLMs typically reason textually from one static mathematical image, while visual CoT can insert redundant signals and lacks hierarchical decomposition for mathematical reasoning.

    Source: §1 · pp. 1–2

  • Method

    ViRC chunks multimodal mathematical reasoning into CRUs and uses CRUX annotations and progressive Instructional SFT, Practice SFT, and Strategic RL training.

    Source: §1 · p. 2

  • Results

    18.8% average performance gain over baselines on multimodal mathematical benchmarks.

    Source: §1 · p. 2

  • Takeaways & Limitations

    VIRC coordinates visual and textual reasoning through structured CRUs, with reported gains across multiple mathematical benchmarks.

    Source: §1 · p. 2, §5 · p. 8

Abstract

from arXiv · show

CoT has significantly enhanced the reasoning ability of LLMs while it faces challenges when extended to multimodal domains, particularly in mathematical tasks. Existing MLLMs typically perform textual reasoning solely from a single static mathematical image, overlooking dynamic visual acquisition during reasoning. In contrast, humans repeatedly examine visual image and employ step-by-step reasoning to prove intermediate propositions. This strategy of decomposing the problem-solving process into key logical nodes adheres to Miller's Law in cognitive science. Inspired by this insight, we propose a ViRC framework for multimodal mathematical tasks, introducing a Reason Chunking mechanism that structures multimodal mathematical CoT into consecutive Critical Reasoning Units (CRUs) to simulate human expert problem-solving patterns. CRUs ensure intra-unit textual coherence for intermediate proposition verification while integrating visual information across units to generate subsequent propositions and support structured reasoning. To this end, we present CRUX dataset by using three visual tools and four reasoning patterns to provide explicitly annotated CRUs across multiple reasoning paths for each mathematical problem. Leveraging the CRUX dataset, we propose a progressive training strategy inspired by human cognitive learning, which includes Instructional SFT, Practice SFT, and Strategic RL, aimed at further strengthening the Reason Chunking ability of the model. The resulting ViRC-7B model achieves a 18.8% average improvement over baselines across multiple mathematical benchmarks. Code is available at https://github.com/Leon-LihongWang/ViRC.

1. Introduction

VIRC addresses the limits of static or indiscriminate visual reasoning by organizing multimodal mathematical CoT into CRUs, with visual information added across units to support later propositions. It pairs this framework with the CRUX dataset and progressive training, and reports gains over baselines.

1. Introduction

  • Existing MLLMs often reason textually from a static image, while visual CoT may inject redundant signals at every step and lack hierarchical decomposition.The cited approaches are also described as constrained to a single reasoning path.Source: §1 · pp. 1–2
  • Human problem-solving selectively examines visual cues and chunks proof steps around intermediate propositions, aligning with Miller’s Law and mathematical assessment’s scoring points.The geometry example combines three image-related proof steps with a proposition-integration step.Source: §1 · p. 2
  • Reason Chunking organizes visual reasoning into Critical Reasoning Units, each validating a self-contained proposition textually while visual information across units supports subsequent propositions.This creates a hierarchy for decomposing complex problems and adapting visual attention.Source: §1 · p. 2
  • CRUX contains 100K samples with one correct and two plausible incorrect paths per sample, four reasoning modes, three visual tools, and CRU-level annotations.Its annotation pipeline samples diverse paths, maps steps to CRUs, then grounds CRUs; the training strategy progresses through Instructional SFT, Practice SFT, and Strategic RL.Source: §1 · p. 2
  • 18.8% average performance gain over baselines on multimodal mathematical benchmarks, with a further 9% average gain on cross-domain high-resolution image benchmarks.The mathematical benchmarks are MathVista, MMStar, and GeoQA; the high-resolution benchmarks are VisualProbe, V*, and HR-Bench.Source: §1 · p. 2

2. Related Work

Prior multimodal mathematical reasoning work improves models through training or curated data, while visual CoT adds visual signals during reasoning. The related work identifies static image encoding and indiscriminate visual integration as limitations.

2. Related Work

  • Existing mathematical-reasoning approaches improve training or curate detailed solutions, yet assume one static image encoding for the reasoning process.The cited approaches include Hint-GRPO, R1-VL [54], Mulberry, and MM-Eureka [23].Source: §2 · p. 3
  • Visual CoT interleaves visual information through tool use, code-executed operations, or imagined feedback to support fine-grained image understanding.Examples include DeepEyes, Mini-o3 [18], OpenThinkImg, and Visual Planning.Source: §2 · p. 3
  • Most VCoT methods still add visual tokens at every reasoning step, increasing computational overhead and introducing redundant or weakly relevant signals.This contrasts with consulting visual information only when necessary.Source: §2 · p. 3

3. VIRC

VIRC structures multimodal mathematical reasoning into proposition-level Critical Reasoning Units (CRUs), builds CRUX to supervise that structure, and trains the model through staged text, grounded multimodal, and reinforcement-learning phases.

3.1. Notations and Preliminaries

  • The notation models an MLLM response to an image and question, distinguishing text-only CoT from VCoT’s visual-token and reasoning-step alternation.CoT uses global visual context at the start, whereas VCoT interleaves visual tokens before each reasoning step.Source: §3.1 · pp. 3–4

3.2. Reason Chunking

  • Each CRU combines dynamically introduced visual tokens with one or more coherent textual steps that jointly verify a complete proposition.Reason Chunking supports on-demand visual invocation and hierarchical decomposition; CoT is a single-CRU case, while VCoT makes each reasoning step its own CRU.Source: §3.2 · p. 4

3.3. Dataset Curation

  • CRUX expands MINT-CoT’s 54K problems [6] into 100K CRU-aligned instances through diverse-path sampling, proposition grouping, and visual grounding.Its paths use planning, reflecting, verifying, and backtracking patterns, operationalized as tool-call transformations aligned with CRUs.Source: §3.3 · p. 5

3.4. Training strategy

  • Training progresses from text-only CRU-structure SFT to grounded multimodal SFT and Strategic RL over a curated hard subset.RL rewards answer correctness, multimodal coherence, reasoning-pattern alignment, and valid formatting; assigning rewards at the CRU level avoids noisy step-level supervision.Source: §3.4 · pp. 5–6

4. Experiment

VIRC-7B outperforms the reported baselines on multimodal mathematical benchmarks and generalizes to high-resolution vision benchmarks. Ablations examine the CRU, reasoning patterns, training stages, and Strategic RL components.

4.1. Experimental Settings

  • Evaluation covers GeoQA, MathVista-Math, and MMStar-Math, alongside high-resolution vision benchmarks VisualProbe, V*, and HR-Bench.GeoQA uses the Geo170K test set; the high-resolution benchmarks span images from 2K to 16K.Source: §4.1 · p. 6, §4.3 · p. 7

4.2. Quantitative Results

  • 18.80% average accuracy gain over Qwen2.5-VL-7B-Instruct and 7.44% average accuracy over MM-Eureka; VIRC-7B tops all three mathematical benchmarks.Compared with the best prior result on each benchmark, VIRC-7B gains +7.7% on GeoQA, +1.2% on MMStar-Math, and +5.18% on MathVista-Math.Source: §4.2 · pp. 6–7
  • On VisualProbe, V*, and HR-Bench, VIRC shows competitive cross-domain performance and robust generalization across all three high-resolution benchmarks.The benchmarks span image resolutions from 2K to 16K; Table 2 identifies the cross-domain test sets and marks results from [18].Source: §4.1 · p. 6, §4.2 · p. 7, Table 2 · p. 7

4.3. Ablation Study

  • Without CRUs, models must learn chunk boundaries and visual-evidence use together, tending toward step-wise Visual CoT with redundant or ambiguous visual signals.The caption describes CRUs as establishing logical hierarchies for tool calls and substantially outperforming step-wise Visual CoT on MathVista-Math.Source: §4.3 · p. 7, Table 3 · p. 7
  • Planning provides a strong baseline, but isolated backtracking, reflecting, or verifying additions can underperform it; integrating all four patterns yields the strongest, more robust framework.The authors attribute this to possible conflicts between a specialized local strategy and the broader problem-solving demands.Source: §4.3 · p. 7
  • Instructional SFT alone improves GeoQA by 26.77% and MathVista-Math by 9.63%; performance continues improving across most benchmarks, peaking after all three stages.Strategic RL provides the largest gains on high-resolution tasks.Source: §4.3 · pp. 7–8, Table 5 · p. 8
  • Removing the multimodal reward prompts answer guessing without tools, while removing the hard subset or pattern-alignment reward drives excessive tool use and fewer final answers.The full Strategic RL configuration achieves the strongest performance and supports exploration and selection of image-grounded reasoning paths.Source: §4.3 · p. 8, Table 6 · p. 8

5. Conclusion

VIRC structures visual mathematical reasoning into CRUs, supported by CRUX and a three-stage training pipeline. Experiments report gains on multimodal mathematics and cross-domain high-resolution image benchmarks.

5. Conclusion

  • VIRC uses Reason Chunking to decompose complex problem-solving into CRUs for dynamic visual verification and coherent validation of intermediate propositions.CRUX supplies explicit CRU annotations and diverse reasoning paths, and the training pipeline comprises Instructional SFT, Practice SFT, and Strategic RL.Source: §5 · p. 8

Overview

The supplementary background connects Miller’s Law to chunked mathematical and visual reasoning, then describes the CRUX dataset section.

A. Background on Miller’s Law

  • Miller’s Law motivates grouping low-level information into meaningful chunks, which the background relates to reduced cognitive load and clearer reasoning.For mathematical and visual reasoning, it describes forming intermediate propositions and selectively attending to relevant visual cues.Source: A. Background on Miller’s Law · p. 12
  • VIRC operationalizes chunking as a hierarchical sequence of CRUs, each a semantically complete unit containing one intermediate proposition.Selective visual grounding occurs between CRUs rather than continuously.Source: A. Background on Miller’s Law · pp. 12–13

B. Dataset Details

  • The CRUX dataset section covers dataset statistics, automated curation of reasoning paths and CRUs, and modeling four cognitive reasoning patterns.The patterns are Planning, Reflecting, Verifying, and Backtracking.Source: B. Dataset Details · p. 13

B.1. Dataset Statistic

This appendix section outlines CRUX statistics, automated construction of CRU-aligned paths, and rule-based edits that instantiate four reasoning patterns.

B.1. Dataset Statistic

  • CRUX statistics cover total data points, CRUs per sample, and the frequency of tool calls and reasoning patterns.The section directs readers to Table 7 for the dataset’s key statistics.Source: §B.1 · p. 13, Table 7 · p. 13

B.2. Dataset Curation

  • Curation samples correct and incorrect paths across image scales, aligns their steps to CRUs, retains steps through first errors, and grounds each CRU to an image region.The paths are composed in the order p1 → p2 → p0; the Linker associates each CRU with a crop tool call.Source: §B.2 · p. 13

B.3. Modeling Cognitive Reasoning Patterns

  • Rule-based postprocessing instantiates planning, reflecting, verifying, and backtracking through edits to the initial tool invocations.Planning adds global context; reflecting reuses a prior crop, verifying displays the original image at the first error in p2, and backtracking upscales the image at the first error in p1.Source: §B.3 · pp. 13–14

C. Additional Implementation Details

VIRC-7B and VIRC-3B use the same three-stage training pipeline, with Strategic RL operating on truncated CRUX reasoning fragments. The supplementary details specify stage procedures, hard-subset construction, and reward components.

C. Additional Implementation Details

  • Both model sizes use Instructional SFT, Practice SFT, and Strategic RL; the vision encoder and vision-language projector remain frozen.The two variants share training configurations and hyperparameters.Source: C. Additional Implementation Details · p. 14

C.1. Instructional SFT

  • Instructional SFT uses 50K text-only CRUX samples with tool outputs removed, preserving only textual supervision.It runs for three epochs at a 5.0e-6 learning rate with batch size 64.Source: §C.1 · p. 14

C.2. Practice SFT

  • Practice SFT uses the same 50K samples, executes tool calls offline, and inserts cached returned images after each call.The stage removes the prompt’s “# Emergency” section so the model uses visual cues during training.Source: §C.2 · p. 14

C.3. Strategic RL

  • Strategic RL applies GRPO to truncated reasoning fragments from a curated hard subset, rewarding answer correctness, multimodal coherence, reasoning-pattern alignment, and format validity.The hard subset is designed to make the model either answer with sufficient context or continue tool-guided reasoning from partial evidence.Source: §C.3 · p. 14

C.4. Hard Subset Curation

  • The 8K-example RL hard subset splits evenly between long-reasoning truncations and critical-region truncations, testing information integration and tool selection.Long-reasoning cases remove the final CRU; critical-region cases truncate at a small, late tool-returned image region.Source: §C.4 · pp. 14–15

C.5. Reward Design

  • The multimodal reward combines textual coherence and visual relevance with pattern alignment, balancing component maxima against answer reward to prevent reward hacking.Textual coherence and visual relevance receive weights of 0.4 and 0.5, respectively.Source: §C.5 · p. 15

D. Evaluation Protocol

The evaluation enables each baseline’s official or built-in visual tools for a fair comparison and uses a separate model to check answer correctness. The supplementary-results section adds detailed quantitative studies, ablations, and qualitative examples.

D. Evaluation Protocol

  • Baselines with official or built-in visual tools are evaluated with those tools enabled, while MINT-CoT uses its official interleaved visual-token mechanism.This unified setting is intended to let baselines fully exploit their visual reasoning capabilities.Source: D. Evaluation Protocol · p. 15
  • Qwen2.5-72B-Instruct verifies the correctness of model-generated answers.The validation prompt is provided in Fig. 10.Source: D. Evaluation Protocol · p. 15

E. Additional Results

  • The additional results section provides detailed quantitative results, controlled studies, fine-grained ablation data, and qualitative reasoning examples.The ablation results break out subset- and category-level metrics.Source: E. Additional Results · p. 15

E.1. Additional Quantitative Results

Across additional benchmarks and settings, VIRC reports cross-domain generalization, consistent gains across pretrained model families, and performance above MINT-CoT and the teacher on multiple benchmarks. It also reports low average inference cost while retaining competitive worst-case cost.

Generalization under domain shift

  • VIRC generalizes from CRUX’s low-resolution schematic diagrams to 2K–16K natural images and perception-centric questions without domain-specific fine-tuning.The evaluation covers VisualProbe, V*, and HR-Bench.Source: §E.1 · p. 15, Table 8 · p. 16

Robustness to pretrained models

  • VIRC training yields consistent improvements with different pretrained model families, including fixed-resolution Qwen2-VL and dynamic-resolution Qwen2.5-VL.The supplied results include VIRC-Qwen2VL-2B and VIRC-Qwen2VL-7B.Source: §E.1 · p. 15, Table 9 · p. 16

Benefits beyond distillation from the teacher model

  • VIRC-7B surpasses the Qwen2.5-VL-72B-Instruct teacher on multiple benchmarks.The authors attribute this to chunk-level organization and denser training signals from reason chunking.Source: §E.1 · p. 15, Table 10 · p. 16

Broader multimodal math benchmarks

  • VIRC-7B consistently outperforms MINT-CoT across MathVision, We-Math, MathVerse, and LogicVista, where MINT-CoT is within 1% of the main-paper best model.The comparison includes MINT-CoT as a multimodal-CoT baseline.Source: §E.1 · p. 15, Table 11 · p. 16

Inference-time cost

  • VIRC-7B has the lowest average token usage and inference latency, and the second-best worst-case cost; removing CRU increases both average measures.Worst-case settings use the top 1% longest token sequences and slowest 1% inference cases.Source: §E.1 · p. 15

E.2. Detailed Ablation Study Results

The supplementary ablation section provides fine-grained results and describes controlled comparisons of CRU guidance, reasoning patterns, training stages, and RL components. A GeoQA example contrasts VIRC-7B’s correct answer with GPT-4o’s incorrect answer.

E.2. Detailed Ablation Study Results

  • Tables 13–16 report subset- and category-level ablation results for CRUs, reasoning patterns, the three training stages, and RL components.The supplied captions identify the table scopes but provide no cell values.Source: §E.2 · p. 16, Table 13 · p. 17, Table 14 · p. 17
  • Ablations hold the base model and training conditions constant except for the component being changed, isolating the impact of CRU guidance, reasoning patterns, training stages, and RL components.CRU guidance is tested by removing Instructional SFT; RL ablations replace the curated subset or remove one reward component.Source: §E.2 · pp. 17–18

Qualitative Comparison on the GeoQA Benchmark

  • In the GeoQA square-area problem, VIRC-7B derives a minimum area of 8 by expressing the area as x^2 + 8 and minimizing at x = 0.The reasoning uses the tangent-radius right triangle and identifies the square’s side with PQ.Source: Qualitative Comparison on the GeoQA… · pp. 19–20

Qualitative Comparison on the MathVista-MINI Benchmark

The MathVista-MINI examples contrast a VIRC-7B solution ending in answer C with a GPT-4o solution ending in answer B. The VIRC-7B trace includes image operations and successive revisions of its reasoning.

VIRC-7B Final Answer: C (Correct)

  • VIRC-7B checks triangle inequalities, derives 5.5 < z < 13, and identifies z = 12 as a choice within that interval.The trace first tests z = 12, then uses a crop and further checks before returning option C.Source: Qualitative Comparison on the… · pp. 22–23

GPT-4o Final Answer: B (Incorrect)

Examples in VisualProbe Hard Benchmark

The examples and prompt templates show visual localization and reasoning support alongside checks for answer consistency, step alignment, and reasoning quality.

Examples in VisualProbe Hard Benchmark

  • In the VisualProbe example, the model crops the region containing the man in a bowler hat and identifies his tie as black.The prompt asks for the tie color; the trace first asks where to look, then examines the cropped image.Source: Examples in VisualProbe Hard Benchmark · p. 24

F.2. Prompts for Data Curation

  • The data-curation prompt samples a reasoning path from an image–text pair using zero-shot chain-of-thought prompting.Figure 9 describes the sampled output as a reasoning path.Source: Figure 9 · p. 28

Answer Validation Prompt Template (Sampling Stage)

  • The answer-validation prompt judges whether the model answer matches the standard answer in meaning, without evaluating the solution process.It outputs 1 for consistent answers and 0 for different answers.Source: Answer Validation Prompt Template… · p. 29

Proposition-Level Reasoning Alignment Prompt Template (Mapping Stage)

  • The reasoning-alignment prompt maps steps by shared purpose and identifies the earliest foundational error in the incorrect path.Its procedure traces backward to distinguish an original error from a later error that depends on it.Source: Proposition-Level Reasoning Alignment… · pp. 30, 32

Planning Pattern Generation Prompt Template (Grounding Stage)

  • The planning prompt requests an image caption and an actionable rationale for applying a mathematical principle, without solving or giving the final answer.The rationale should specify how the principle applies to diagram elements and direct the first logical step.Source: Planning Pattern Generation Prompt… · p. 32

Bounding Box Localization Prompt Template (Grounding Stage)

  • The localization prompt asks the model to locate a geometric structure, identify necessary image text, and return bounding boxes for that text.The three requested operations are structure recognition, auxiliary recognition, and text detection.Source: Bounding Box Localization Prompt… · pp. 33–34

F.4. Prompts for Reinforcement Learning

  • The reinforcement-learning prompts evaluate answer correctness, textual reasoning quality, and multimodal reasoning quality using the current step and relevant context.The textual evaluator considers preceding steps, while the multimodal evaluator also considers the image and tool results.Source: §F.4 · p. 36, Figure 18 · p. 36, Figure 19 · p. 37

A CRU-aligned reasoning path in the CRUX dataset

The example presents a geometry problem about finding EF in a square, with a reasoning trace that uses image crops and repeatedly revises its geometric approach.

A CRU-aligned reasoning path in the CRUX dataset

  • The example asks for EF in a square with AE = 3 and CF = 1, using perpendiculars from the diagonals’ intersection to two sides.The question offers four answer choices, including √10 and 2√2.Source: A CRU-aligned reasoning path in the… · p. 26
  • The trace first treats AEO and COF as right triangles, then crops the image and revises its coordinate-based reasoning about E and F.It also tries scaling the image and later displays another crop.Source: A CRU-aligned reasoning path in the… · pp. 26–27
  • The figure caption identifies the example as a CRU-aligned reasoning path in the CRUX dataset.The excerpted trace does not include a final answer to the EF question.Source: Figure 8 · p. 28
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