CogFlow: Bridging Perception and Reasoning through Knowledge Internalization for Visual Mathematical Problem Solving
Authors: Shuhang Chen
Yunqiu Xu
Junjie Xie
Aojun Lu
Tao Feng
ZEYING HUANG
ZHANG NING
Yi Sun
Yi Yang
Hangjie Yuan
Venue: ICLR 2026 Poster (2026)
Keywords: CogFlow
MLLM
RLHF
visual mathematics problem
Primary area: applications to computer vision, audio, language, and other modalities

Abstract: Despite significant progress, multimodal large language models continue to struggle with visual mathematical problem solving.
Some recent works recognize that visual perception is a bottleneck in visual mathematical reasoning, but their solutions are limited to improving the extraction and interpretation of visual inputs.
Notably, they all ignore the key issue of whether the extracted visual cues are faithfully integrated and properly utilized in subsequent reasoning.
Motivated by this, we present CogFlow, a novel cognitive-inspired three-stage framework that incorporates a knowledge internalization stage, explicitly simulating the hierarchical flow of human reasoning: perception$\Rightarrow$internalization$\Rightarrow$reasoning.
In line with this hierarchical flow, we holistically enhance all its stages.
We devise Synergistic Visual Rewards to boost perception capabilities in parametric and semantic spaces, jointly improving visual information extraction from symbols and diagrams.
To guarantee faithful integration of extracted visual cues into subsequent reasoning, we introduce a Knowledge Internalization Reward model in the internalization stage, bridging perception and reasoning.
Moreover, we design a Visual-Gated Policy Optimization algorithm to further enforce the reasoning is grounded with the visual knowledge, preventing models seeking shortcuts that appear coherent but are visually ungrounded reasoning chains.
Moreover, we contribute a new dataset MathCog for model training, which contains samples with over 120K high-quality perception-reasoning aligned annotations.
Comprehensive experiments and analysis on commonly used visual mathematical reasoning benchmarks validate the superiority of the proposed CogFlow.
Project page: https://shchen233.github.io/cogflow.

Published as a conference paper at ICLR 2026
COGFLOW: BRIDGING PERCEPTION AND REASONING THROUGH KNOWLEDGE INTERNALIZATION FOR VI-
SUAL MATHEMATICAL PROBLEM SOLVING
Shuhang Chen' Yunqiu Xu² Junjie Xie
Aojun Lu Tao Feng"
Zeying Huang² Ning Zhang Sun Yi Yang! Hangjie Yuan
'Zhejiang University Intelligent Learning Sichhunn University TTinggua University (sh. chen, jj. xie, yangyics, hj.yuanneezu.eedu..
(imyunqiuxu, fengtao.hi futuretrader)egmall aojunluestu, scu, en jzxjeffe163 com
suny18jex100. 00000.000..0 cn
ABSTRACT
Despite significant progress, multimodal large language models continue strug- gle with visual mathematical problem solving. Some recent works recognize that visual perception is bottleneck in visual mathematical reasoning, but their lutions are limited to improving the extraction and interpretation visual inputs. Notably, they all ignore the key issue of whether the extracted visual cues are faithfully integrated and properly utilized in subsequent reasoning. Motivated by this, present COGFLOOW, novel cognitive-inspired three-stage framework that incorporates knowledge internalization stage, explicitly simulating the hi- erarchical flow of human reasoning:
line with this hierarchical flow, we holistically enhance all its stages. We devise Synergistic Visual Rewards boost perception capabilities in parametric and se- mantic spaces, jointly improving visual information extraction from symbols and diagrams. To guarantee faithful integration extracted visual cues into subse- quent reasoning, we introduce Knowledge Internalization Reward model in the internalization stage, bridging perception and reasoning. Morcover, we design Visual-Gater Policy Optimization algorithm to further enforce the reasoning is grounded with the visual knowledge, preventing models seeking shortcuts that appear coherent but visually ungrounded reasoning chains. Moreover, con- tribute a new dataset MATICCOG for model training, which contains samples with over 120K high-quality perception-reasoning aligned annotations. Comprehen- sive experiments and analysis on commonly used visual mathematical reasoning benchmarks validate the superiority of the proposed COGFLOW, Project page: https: /tttp:://shchn2333 wwt.tiithhh...... 10/ coggollw/
INTRODUCTION
Multimodal large language models (MLLMs) are rapidly advancing and have been applied across
various vision language applications (Peng et al. 2024b|
Xu et al.) [224 2025b Yue et al. 2024 2024 Zhang et al.] 2025c Ma et al. 2024a b)
Liang et al. 2025 Jia et al. 2024a Quan et al.
However, existing MLLMS continue struggle with challenging visual mathematical problems,
sulting in low answer accuracy and inconsistent reasoning chains. Some early attempts (Wang et al.) 2025a Shen al., 2025) adopt one-step reasoning framework that directly interleaves visual per- ception with reasoning in an unstructured manner, often resulting in both perceptual and reasoning errors. Another line of work (Chen et al.) 2025a] Guo et al.) (202cc) follows a decoupled reasoning pipeline that explicitly separates the perception and reasoning parts. with the former focusing on subsequent inference. Yet practice, we observe
visual recognition and the latter responsible that such a pipeline often suffers from the reasoning drift issue, i.e., it tends to yield illogical or unwarranted reasoning steps that disregard perceptual evidence (see Figure TT. These observations
"Corresponding author.

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Published as a conference paper at ICLR 2026
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MathFlow CoGFLoW (b) Reasoning drift analysis across three representative pipelines, where higher precision indicates less reasoning drift.
VLM-R1
(a) One-step framework yields unstructured reasoning, while decou- pled pipeline modularly disentangles the flow. We adopt a cognitive- inspired three-stage framework with knowledge internalization.
The one-step reasoning framework (e.g., VLM-R1 (Shen et al., 2025 often yields sub- optimal results, while the decoupled reasoning pipeline (e.g.. MathFlow (Chen et al.] (025a)) en- hances perception yet still yields illogical reasoning steps that disregard visual evidence. In contrast, COGFLOW adopts cognitive-inspired three-stage framework to effectively mitigate reasoning drift.
Figure 1:
motivate the development of a new approach that not only achieves robust fine-grained recogni- tion of mathematical visual elements (e.g., diagrammatic primitives and symbols) but also faithfully incorporates extracted visual cues into subsequent reasoning.
Inspired by cognitive science findings knowledge internalization (Ryan & Connell 1989 Landy ct al. 2014, Wu et al./ 2022), this paper introduces COGFLOW, a novel three-stage visual mathe- matical reasoning framework that better mirrors the typical hierarchical structure of the hu- man reasoning process. Concretely, after perception captures raw sensory input, an intermediate knowledge internalization stage transforms low-level perceptual signals into structured and semanti- cally grounded knowledge representations (e.g., humans internalize the perceptual facts that the line segment AB is diameter and the point C lies the circle into the knowledge that ZACB 90%) before high-level reasoning begins. As illustrated in Figure Ia to ensure both accurate extraction of visual information and its faithful use reasoning, COGFLOW explicitly models the hierarchical sequence of the human reasoning flow (i.e., Operception @internalization aamaliiattion-rrroonnnnn and holistically enhances all three stages in synchrony with it, where each improvement tailored to the functional role of the corresponding stage in the human reasoning process.
Unlike prior approaches (Jia et al., 2024b] Chen al., 2025a Guo et al.) (025c) that decouple perception from reasoning trajectories (Ouyang et al., 2022) and enhance it with tailored tasks, COGFLOW first integrates perception enhancement into a unified reinforcement learning (RL) framework through Synergistic Visual Rewards (SynVRs), enabling dynamic percep- tion-reasoning interaction and improving generalization. Specifically, Syn complementarily optimize the model from two distinct perspectives: (1) Visual Parameterized Reward (VPR) en- coding normalized primitives (i.e., points, lines, and circles) and calculating the Euclidean distance in a parameter space for precise and interpretable measurement; (2) a Visual Semantic Reward (VSR) that extracts semantic embeddings (Xie et al. (025) from re-rendered images (derived from textual perception outputs) and measures the cosine distance in semantic space capture holistic style and layout consistency. Together, VRs ensure both local geometric fidelity and global per- ceptual coherence, forming trustworthy visual cues that serve as the foundation for effective visual mathematical reasoning.
Notably, despite progress in perception enhancement, all prior efforts (Jia et al. 2024b Guo et al., 2025c Wei et al. 2025) remain confined to accurate extraction of mathematical information from diagrams, while ignoring a key question: are the extracted visual cues properly and faithfully inte- grated into subsequent reasoning? ? As illustrated in Figure lb our empirical findings reveal typical reasoning drift issue (i.e., the reasoning stage in existing methods often deviates from perceptual results), leading to reasoning chains that appear coherent yet conflict with the underlying visual evi- dence. To prevent such drift and improve interpretability, COGFLOW utilizes a Knowledge Inter- nalization Reward (IntlzR) that bridges the perception and reasoning stages by encouraging
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Published as a conference paper at ICLR 2026
the model to generate structured and reasoning-ready outputs (i.e., knowledge-nnternalied representations (Ryan & Connell 1989) as more reliable foundation for subsequent reasoning. Specifically, we curate positive trajectories integrating perception and reasoning processes with ex- plicit internalization of perception primitives, and further derive five typical negative trajectories. Training with these trajectories enables the reward model to evaluate each response according to its fidelity to the internalized representation. IntlR effectively improves the knowledge internalization stage, thereby reducing hallucinations and improving interpretability and robustness.
In accordance with the hierarchical flow of human reasoning, we further improve multi-step visual reasoning beyond enhanced perception and knowledge internalization. Existing approaches either follow a text-centric RL paradigm (Guo et al.) (025a) that is free from perceptual objectives (Chen et al. 2025b, Wang et al.) 2025b), or overlook the structured dependency between perception and reasoning (Shen et al. [025] Wang et al. (025a). To ensure more stable reasoning in the presence of perceptual errors, COGFLOW introduces Visual-Gated Policy Optimization (VGPO) strategy that explicitly anchors the reasoning process in perceptual accuracy. In VGPO, visual gate is designed to adaptively filter perceptual trajectories through perceptual quality assessment, retaining only high-quality ones before subsequent reasoning trajectory generation. If a low quality perceptual trajectory is filtered out, the model regenerates alternative trajectories to obtain a higher-quality response. Along with the proposed visual gate, VGPO integrates an outcome-supervised Inference Reward (Shao et al. 2024) for optimization, further strengthening multi-step visual reasoning.
To facilitate research, we curate subsets and over 120K samples with high-quality perception-reasoning aligned annotations. We conduct extensive experiments on commonly used visual math problem-solving benchmarks (Chen) et al. 2025a Zhang et al.) 2024] Lu et al. 2024, Qiao et al. 2022a; Xiao et al.) 2024 Zou el al. 2025 to comprehensively evaluate COGFLOW. The results show that COGFLOW consistently out- performs state-of-ine ofte--ff-heearrr the-art MLLMs with comparable model sizes. Notably, it achieves on-par or even better results compared to advanced closed-source MILLMs with much larger model sizes. The main contributions of this paper can be summarized as follows:
MATHCOG dataset for model training, which contains three
All prior works neglect whether extracted visual cues are faithfully used in reasoning. To address this issue, we present COGFLOW, a novel cognitive-inspred three-stage framework that faithfully simulates the hierarchical human reasoning flow: inrcpptooniinnemmalizziioooo
In line with human reasoning hierarchy, COGFLOW holistically enhances all three stages: Syn complementarily enhance accurate and complete diagram perception in parametric and semantic spaces; IntlzR improves the knowledge internalization ability for promoting faithful conversion of perceptual outputs into a canonical context used for subsequent inference; VGPO employs a visual gate filter high-quality perception trajectories and enhances the stability of reasoning.
To support model training, we curate new dataset MATHCOG with disentangled perception and reasoning annotations. Comprehensive experiments on multiple visual mathematical benchmarks validate that CoGFLOW achieves substantial gains in both answer accuracy and reasoning quality.
2 RELATED WORK
Visual Mathematical Reasoning with MLLMs. Solving visual mathematical problems (e.g., ge- ometry diagrams, algebraic plots, and etc) requires both strong reasoning ability and accurate in- terpretation of visual primitives and symbolic content (Yan al. 2025 Qiao et al. 2025b Zhang
et al.) 2025b). Most previous works are dedicated to improving the reasoning process, including
chain-of-thoughe strategies (Xu et al. 2025a Deng et al., 2024). tool-aided reasoning (Trinh et al.)
2024 Chen et al.] 2023b)) test time scaling WWag et al. 2025e Hosseini et al. 2024), and rein- forcement learning ( (Wang et al. 2025c Jiang et al. 2025). Several recent works (Guo et al., 2025c Jia et al.) 2024b: Wei et al. 2024) suggest that one of the major bottlenecks in visual mathematical reasoning is inaccurate visual comprehension. They typically decouple perception from reasoning, and strengthen perception either by designing specialized visual encoders (Zhang et al. 2025a) or by introducing auxiliary visual tasks (Chen et al.] 2025a).. However, prior works ignore a key issue of whether correctly extracted visual cues are indeed faithfully incorporated into subsequent reasoning.
Reinforcement Learning for Multimodal Reasoning. Traditional actor critic methods, such as proximal policy optimization (Yu et al. (022), are computationally expensive. A lightweight alter-
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Visual-Gated Policy Optimization
Input Data
Gradient Update
Visual Gate
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Rewards
Group Computation
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Figure 2: Overview of the proposed visual mathematical reasoning framework COGFLOW. In- spired by the canonical three-stage human reasoning flow, COGFLOW adopts hierarchical pipeline that integrates Synergistic Visual Rewards (SynVRs) for enhanced perception, a Knowledge Inter- nalization Reward (IntIzR) to bridge perception and reasoning, and Visual-Gated Policy Optimiza- tion (VGPO) with Inference Reward (InfR) to anchor multi-step reasoning in perceptual accuracy.
native is group relative policy optimization (GRPO) (Guo et al. 2025a which stabilizes advantage
estimation using group baselines. While its variants (Zheng et al. |025 Yu et al. (2025) have been widely explored, GRPO has also been extended to multimodal reasoning (Wang et al. 2025c Shen
al. 2025 Zhou al., 2025). Some extensions introduce hybrid reward formulations augmented with preference signals during training (Wang et al. 2025c Pan et al. (2025). whereas others propose two-stage multimodal reinforcement learning paradigms, e.g., OVR (Wei al. 2025). However, ex- isting methods typically lack explicit mechanisms to strengthen the alignment between perception and reasoning, often leading to reasoning that is not firmly grounded in visual content.
COGFLOW: A COGNITIVE- INSPIRED HIERARCHICAL FRAMEWORK
Inspired by the typical cognitive process of human reasoning (i.e., 1 perceetioon @internalization @reasoning), COGFLOW serves as a visual mathematical reinforcement learning framework that explicitly implements the internalization stage (see Figure
Before training, we first curate MATHCOG to support the subsequent training (see Figure TO Specifically, the training pipeline of COGFL.OW consists of two sequential phases: a supervised fine-tuning (SFT) phase and a reinforce- ment learning (RL) phase. The SFT phase endows the base model with initial visual perception and basic reasoning skills based on the MATHCOG- SFT dataset. During the RL phase, we optimize the policy based the MATHCOG-RE RL dataset under the Visual-Gated Policy Optimization (VGPO) framework to explicitly anchor the reasoning process in perceptual accuracy. Concretely, VGPO introduces visual gate adaptively filter perceptual trajectories before reasoning trajectory gen- eration. Furthermore, the rewards in VGPO are composed of three components: Synergistic Visual Rewards (Syn VRs) for forming trustworthy perception, Knowledge Internalization Reward (IntlzR)
for detecting reasoning drift and Inference Reward (InfR) for providing outcome-supervision
3.1 FORMING TRUSTWORTHY PERCEPTION WITH SYNERGISTIC VISUAL REWARDS
COGFLOW first constructs Synergistic Visual Rewards (SynVRs), enabling dynamic perception- reasoning interaction and improving generalization. Specifically, the proposed SynVRs combine two complementary components: the Visual Parameterized Reward (VPR) and the Visual Seman- tic Reward (VSR), which respectively evaluate perceptual quality in the parametric and semantic spaces, thereby providing a synergistic perception feedback integrated into RL training loops.
Measuring Perceptual Accuracy in Parameter Space. As shown in Figure 3 VPR first con- verts structured visual information into parametric expressions, For example, the primitive Circle (0.01, 0.46, 7.68) is transformed into the Equation (r 0.01)² + (y + 0.46)2 7.68?.
We then compute the cost matrix C between GT Primitives & and predicted Primitives P and apply the Hungarian matching algorithm (Kuhn 1955) obtain the optimal one-to-one matching H. that
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Structured Visual Information
Semantic Space
#T E
Vinua Semenhic
Synengashe
Youee
Enteraal astiion
Krowledge
Infonence
Reward
Liw
Tool
Palicc Mosill
Visual Gotte
Adbertoyy
Leceti
Voua Reword Suya
Campurt
farenatric Exrexsior
Av
Parametric Space
Geadiest Update
Figure 3: Workflow of SynVRs. Syn consist of a Vi- Figure 4: Pipeline of VGPO. VGPO sual Semantic Reward and Visual Parametric Reward, en- suring local geometric fidelity and global perceptual coher- wards to strengthen multi-step visual ence respectively. Together, these two complementary vi- reasoning. Coupling perceptual quality sual rewards provide unified supervision mechanism for control with outcome-based optimiza- training robust and accurate visual perception.
introduces visual gate and multiple
tion promotes stability.
minimizes the total cost SyPR- The VPR offers interpretable, geometry-awant semantic supervision that avoids the pitfalls of pixel-level noise or black-box embedding similarity. More details are provided in Appendix C.2
Layout and Style Consistency in Semantic Space. We render the predicted response into an image and compare it against the ground-truth rendering 7, using a frozen FG- CLIP encoder (Xie et al., (2025) o(-) with higher sensitivity to spatial features. The VSR score SysR is the normalized cosine similarity. The higher values indicate closer agreement in global layout and style while preserving fine-grained geometric fidelity.
Capturing Holistic
Synergy of Complementary Visual Rewards. Finally, the Syn VRs score can be formulated as
Σ (LJ)) ((P..G)) (1 a) 1 (os()()), e(T))
SBynVRs exp
2
Syrr
SVer
where a E [0, 1] balances local geometric precision (SVPR) and global visual consistency (SVaR). SRY has two roles: (1) acts as a gate to prevent low-quality perceptions from propagating during policy generation, and (2) as a training reward during RL to encourage accurate perception.
3.2 ALLEVIATING REASONING DRIFT WITH KNOWLEDGE INTERNALIZATION REWARD
Failure Modes in Knowledge Internalization. Even with accurate perception, models often strug- gle to reliably internalize what they see. The empirical evidence suggests that such failures typically manifest in five forms: (1) omit or mishind primitives, losing essential elements confusing their identifiers; (2) introduce nonexistent facts, fabricating geometric relations that are not present in the perceptual output; (3) invoke external theorems inappropriately, applying mathematical results that were never justified by the internalized structure; (4) contradict geometric constraints, producing inference steps that violate fundamental properties of the canonical representation; and (5) refer inconsistently to established elements, assigning shifting roles or properties to the same primitive across different reasoning steps. Therefore, enhance knowledge internlizatiin, it is crucial to mitigate these systematic failures.
Rewarding Faithful Knowledge Internalization. To address these systematic weaknesses, we introduce the Knowledge Internalization Reward (IntIzR), a trained reward model that evaluates whether each reasoning chain remains faithful to the internalized visual representation, thereby enforcing raw perception as the primary substrate of inference (e.g., reasoning drift). Within COGFLOW, we train the IntlzR to enhance internalization explicitly, supported by the curated MATHCoG-IntlR (see Appendix $C.33 for more details). This dataset is constructed from posi- tive-negative pairs, with each positive matched to five corresponding negatives; positives are sam- pled from MATHCOG-SE FF,, and negatives synthesized by injecting the five error types described
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above, providing fine- fine-grained supervision for distinguishing grounded from unanchored reasoning. To better leverage diverse negative signals in IntlzR and adaptively emphasize the most challenging trajectories while improving gradient efficiency and training stability, we adopt Softmax-DPO (Chen
et al. 2024a) for optimization:
M
st) st))
DDo =log o log
exp (5)
( x)) log Teer(y) x)],
j=1
(2)
where s* denotes the score of the preferred trajectory, {s 1-= represent the corresponding scores of the dispreferred trajectories, the KI. penalty coefficient and a(-) denotes the sigmoid func- tion. This formulation contrasts one positive trajectory against multiple negatives simultaneously, thereby providing denser and more informative supervision, implicitly emphasizing hard negatives through softmax weighting, and yielding more stable optimization with stronger robustness to un- seen misalignment patterns. During RL, IntlzR is evaluated stepwise and aggregated across the chain, rewarding trajectories in which the extracted visual cues are faithfully and adequately inte- grated into subsequent reasoning.
3.3 STRENGTHENING MULTI-STEP VISUAL REASONING WITH VGPO
Accurate perception and faithful internalization are necessary but not sufficient: the model must also produce long reasoning chains that are coherent, interpretable, and verifiably grounded. Hence, we introduce Visual-Gated Policy Optimization (VGPO), which integrates a visual gate with group- level optimization to regularize the reasoning process. Concretely,
shown in Figure A for an input question x we sample M candidate trajectories (x) = (v," (I)) y (ar, y," (ar)). Here y," (x) denotes the perception trajectories, i.e., structured parse of perceptual primitives and relations from the diagram, and y (z, y.º (z)) denotes the reasoning trajectories conditioned on æ and y (z).
Visual Gate for Reliable Visually Grounded Reasoning. We introduce visual gate that scores each perception against the visual evidence and forwards only the most faithful parse to reasoning. Specifically, given a perception candidate y (z) and the ground-trute y (z). while I, and I denote their respective renderings, we define the perceptual accuracy score as:
SveR (u,' (I), (x)) SvsR (I, I), training, Svsn [T,, I),
(v (x))
inference.
Subsequently, the visual gate r-)) enforces perception quality by scoring each perception trajectory and accepting the first attempt whose score exceeds a preset threshold. no attempt passes within M trials, the gate returns the attempt with the highest Svis:
minf k M). Svis (vl(v(((=) 2T). M,
(k: Svis())e r) + Ø [k: (y
K
(x)) 2 T) = Ø
(4)
v (x) (( - = arg max (v),
(5)
ISk5*
where K. is the stopping index, defined as the smallest & whose score reaches acceptance threshold T, 82" * (x) is the perception trajectory selected by the visual gate; and V (z) is the k-th perception trajectory for input x.
Stabilizing Multi-Step Reasoning via Visual-Gated Policy Optimization. For each problem, we collect multiple candidate trajectories yı(x) and evaluate them under group-level reward that integrates (1) the Synergistic Visual Rewards RSynVRs for perceptual fidelity, (2) the Knowledge Internalization Reward RIsIZ for internalization faithfulness, and (3) an Inference Reward RER capturing answer correctness and output format, given by:
ASynVRs RSynVRs + Amaaa RIMEE + APaR REIR
(6)
Then, the training optimization objective L is formulated as:
L C=- E, min (0)))) (n) 1 - 1 + +)) + BKI. DKI( ll moar);
(7)

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Published as a conference paper at ICLR 2026
Table Accuracy (%) and Flow Verse-style CoT-E (%) results on Verse.
Centric
Text Limited
Vision Centric
Vision Primary
Model
CoT-E
Ace
CoT-E
Acc
CoT-E
Acc
CoT-E
Ace
CoI-E
Acc
CoT-E
Acc
CoT-E
Acc
Claude-3.5 3.5-Sonnet
55.5
45.1
60.8
52.6
58.7
50.3
64.0
58.3
45.0
25.4
56.5
48.0
48.1
45.2
GPT-4o
56.9
49.7
61.0
56.8
58.7
54.4
62.2
58.2
45.2
30.0
58.6
52.6
54.1
51.0
GPT-VV
64.2
58.7
69.1
57.1
65.0
55.0
72.0
61.4
48.1
30.3
61.8
46.3
42.0
36.7
MathFlow' CPT 4V
64.2
59.5
69.5
58.2
67.2
57.4
71.1
64.1
52.7
47.5
62.1
57.1
60.4
57.0
Gemini-2.5-pro
64.5
56.2
68.3
61.9
66.1
60.8
68.9
64.1
52.1
37.1
65.7
57.9
57.0
54.6
GPT-5
59.3
74.3
73.5
66.7
69.2
53.8
44.7
67.1
60.3
57.5
Math
37.8
29.5
43.8
38.1
40.6
36.7
46.1
40.1
28.8
15.4
39.6
30.3
26.1
23.2
1ntemBV.2.5-88
46.3
40.1
49.2
41.3
40.5
38.4
49.6
42.7
38.4
20.2
41.0
35.9
35.8
33.9
Math-LLaVA-13B
39.3
30.8
45.1
44,4
37.4
36.2
41.7
35.9
37.0
34,2
MultiMath-7B
45.2
35.3
50.6
44.8
49.9
42.9
41.7
22.1
47.2
40.4
39.7
38.8
SVE-Math: Qwen2.5-7B
38.7
53.1
53.4
45.8
44.2
45.8
VI.M-R1-7B
50.7
41.2
59.0
54.2
57.9
49.8
65.5
58.9
36.2
24.5
46. 1
37.8
30.6
26.1
CoGFLoW-7B
66.0
56.2
67.9
58.6
67.3
58.3
68.1
60.9
57.8
42.7
68.2
61.1
66.7
63.5
Table 2: Accuracy (%) and Math Verse-style CoT-E (%) results on festmini set of Math Verse.
All
Text Dominant
Text Lite
Text Only
Vision Intensive Vision Dominant
Vision Only
Model
CoT-E
Acc
CoT-E
Acc
CoT-E
CoT-E
CoT-E
CoT-E
CoT-E
Acc
Qwen-VI. VL.-luss
21.3
11.8
26.0
15.7
21.2
11.1
25.2
14.5
18.5
9.0
19.1
13.0
21.8
10.0
Gemini
35.3
23.5
26.3
34.7
36.8
33,3
22.2
Dween-VL-Max
37.2
25,3
42.8
30.7
37.7
26.1
47.9
28.9
33.6
24.1
35.9
24.1
35.9
21.4
GPT-4V
39,4
63.1
54.7
56.6
51.4
50.8
50.3
31.6
MathFlow GGP--VV
56.7
43.8
65.2
51.1
58.9
46,4
62.1
48.5
53.7
40.3
52.1
37.4
52.5
39.0
SPHINX-MoE:550
25.8
15.6
33,3
22.2
21.9
16.4
40.7
18.3
21.1
14.8
19.6
12.6
18.3
9.1
MieemL.M-XC2-11
25.9
16.5
36,9
22.3
28.3
17.0
42.5
16.5
20.1
15.7
24.4
16.4
19.8
11.0
Mait-LLLVVA118
20.1
22.8
21.8
21.1
19.2
15.4
MultiMath--B
26.9
34,8
30.8
28.1
25.9
15.0
SVE-MMulh-Qwen225-72
31.4
37.6
36.8
34,9
31.5
16.0
DVLR-14B
48.1
54.3
49.0
46.3
47.2
43.8
SophiaVL-R117B
48.8
45.4
43.9
45.1
58.5
51.3
CoGFLOW-7B
53.9
39.5
60.7
41.9
51.2
37.0
52.3
40.1
55.0
42.4
58.7
44.8
44.2
26.3
vi(x)) (x)( val Tw" (x))))
(0)
Toly: (x)
x))
A:
Ti Mgrrop
To.d (x)
group (r) +
where the E.[-] [+] is the empirical average over candidates in the group; the A; is the group-normalin normalzedd advantage: (0) the likelihood ratio w.t. the behavior policy Te clip(-) uses the PPO-style hyperparameter [lgroup and o group are the group mean and standard deviation of ry (with small stabilizer& ε in the denominator); 3K is the KL penalty coefficient and DKL, (Tel denotes the forward KL to frozen reference policy (e.g., the SFT model), implemented as the token/state- averaged KL in practice.
By combining visual gate with group-level optimization, VGPO stabilizes horizoorizon training and encourages the emergence of interpretable chain-of-thoughts. This mechanism ensures that COGFLOW converges toward the desired paradigm: first perceive correctly, then reason correctly.
EXPERIMENTS
4.1 EXPERIMENTAL SETUP
Evaluation Benchmarks and Metrics. We conduct experiments on widely used benchmarks (i.e., Flow Verse (Chen et al., 2025a). Math (Zhang et al.) (024). Math Vista (Lu et al.) 2024). We- Math (Qiao et al. 2025a). Visic (Xiao et al. 2024), and DynaMath (Zou et al. 2025 with various visual mathematical reasoning tasks and different visual and textual complexity. Following common practice, we measure the accuracy (Acc) of the final answer and assess the reasoning ability from the intermediate reasoning process using the chain-of-thooe of-thoughtttttttthh evaluation (CoT-E).
Baselines. We compare our method with wide range of closed-source MLLMs, such as GPT
series (OpenAl 2024 2023 2025), Gemini series (Team et al. 2025). DouBao- 1.5-pro Guo
et al. 2025b), GLM-4.5V (Hong al. 2025), Claude (Anthropic 2024) and Qwen- VL-Plus (Bai
et al.
2023). as well as open-source MLLMs, including InfiMM Math (Han et al, 2025). In-
tern
12..5 (Chen et al. 2024b), InternI.M-XC (Dong et al. 2024). Math-ILLaVA (hi et al.
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Published as a conference paper at ICLR 2026
Table 3: Accuracy (%) results Math
Table 4: COGFLOW shows competitive accuracy (%) results on more visual math benchmarks.
COGFLOW demonstrates consistent superiority.
Model
All FQA GPS MWP TQA VQA
Models
WeMath Logic Vista DynaMath
GPT-4V
49.9 43.1 50.5 57.5
65.2
38.0
Claude-3.7-Sonnet
49.3
39.7
Claude-3.5-Sonnet
67.7
GLM-4.5V
68.8
62.4
53.9
Doubao-pro-1.55
79.5 77.7
88.9
86.0
82.3
62.0
Doubao-1.5-PMy
65.7
64.2
44.9
GPT-5
71.1
70.0
60.9
G-LLaVA-7B
25.1 19.1 48.7
3.6
25.0
28.7
Gemini-2.5-Pro
78.0
56.3
VCAR-7B
33.7 30.9
34.6
38.7
37.3
28.5
SPHINX-Plus
36.7 54.6 16.4
23.1
41.8
43.0
Ovis-8B
27.2
39.4
20.4
SVE-Math-7B
37.4 31.9
53.9
29.0
41.4
30.8
Qwen2,5-VL-8B
44.1
21.0
MultiMath-7B
50.0 40.1
66.8
61.8
50.0
33.0
VL3-8B
37.1
44.1
25.5
SophiaVL-R1177
71.3
73.4
Keye-VL-SB
VL-8B
60.7
54.8
37.3
ThinkLite-VL VL-7B
71.6
Intern Intern VB
57.0
57.3
37.7
VL- "L-Rethinker-75
73.7
GLM-4.1V-99
63.8
60.4
42.5
COGFLOW-78 7B
76.8
70.4
93.1
73.7 86.9 59.3
COGFLOW-7B
64.1
58.1
46.2
2024..
MultiMath-77B (Peng et al. 2024a). SVE-Math (Zhang et al, 2025b), VLM-RI (Shen et al,
2025
MathFlow (Chen et al. 2025a), DVLR (Guo et al. (025c). VCAR (Jia et al.
2024b). VL-
Rethinker (Wang et al. 2025a), ThinkLite-Vi (Wang et al. 2025d) and SPHINX-Mol (Liu et al.
2024a). Since some concurrent works have yet release their models, report the results provided in their original papers, some of which do not include category- performance.
Implementation Details. We initialize COGFLOW with Qwen2.5-VL-77B (Bai al. 2025 and train it on our curated МATHCOG dataset. For cold start, we first train COGFLOW on МATICО-- SFT subset for 2 epochs with learning
of 10-3 and a batch size of 64. Subsequently, COGFLOW is optimized using VGPO on MATHCOG- RL subset for epoch with a learning rate of 1 x 10-6 and batch size of 16. The IntlzR reward model based on Qwen2.5-VL-3B and trained on MATHCOG-Intlak subset for 3 epochs with learning rate of 7 x 10-6 and batch size 64. All our models are trained on 16 NVIDIA A100 GPUs. Please refer to Appendix $C for more details.
4.2 MAIN RESULTS
As shown in Tables 14 COGFLOW achieves 66.0% accuracy on Flow 53.9% on Math Verse, 76.8% on Mathvista, 64.1% accuracy on WeMath and 46.2% on DynaMath, surpassing all open- source baselines by large margins. In terms of Logic Vista, it delivers competitive performance, trailing only GLM-44IV-9B. COGFLOW'S improvements in visual perception are particularly significant, evidenced by its superior performance nates. For example, on Flow Verse it reaches 42.7% on Vision Dense, 55.6% on Vision Primary, and 61.1% Vision Centric. On MathVerse the gains consistent Vision Dominant (42.1 %), and Vision Only (25.7%). These results demonstrate that CoGFL.ON yields more accurate parsing of geometric primitives and relations, leading to stronger percep- tion-reasoning integration. COGFLOW also delivers higher-end-to--nn problem-solving accu- racy across all settings. The largest absolute gains appear visually demanding subsets, where perception
subsets where visual information domi-
Vision Intensive (44.8%),
-encppioo-anchored anchored reasoning is essential. This demonstrates that COGFLOW not only perceives diagrams more accurately also reasons over them more effectively. Figure 19 further shows that CoGFLon FoW produces more structurally accurate and semantically consistent predictions compared to strong baselines. Please refer to Appendix $D.12 for more detailed configurations.
4.3 ABLATION STUDIES
Component Ablation. Table reports the ablation results of COGFLOW. We observe that every component in our framework contributes positively to overall performance, which confirms the ne- cessity of jointly addressing perception, internalization, and reasoning. Among all modules, VGPO emerges as the most influential, as it directly stabilizes long-horizon reasoning through visual gate and group-level optimization, thereby yielding the largest single- module gain.
Analysis on Synergistic Visual Rewards. As shown in Figure 5 the model without any visual reward demonstrates reasonable performance. However, adding either VSR VPR consistently
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Published as a conference paper at ICLR 2026
Table 5: Ablation of three proposed compo-
nents (i.e,, SynVRs, IntlzR and VGPO). The vi- sual gate always enabled during inference.
FlowVerse CoT-E (%)
MathiVerse CoT-E
67
53.9
-54
66.0
Verse
Math Verse CoT-E
52.8
SynVRs IntlæR VGPO
CoT-E
Acc
Acc
65.2
65
64.5
51.7
-52
x
x
57.4
48.7
48.2
35.6
x
x
63.2
54.7
50.5
36.9
64
63.8
50.9
-51
62.7
53.5
49.9
36.2
X
v
63.4
54.8
50.8
37.3
63
VSR only VPR only
Both
50
x
64.4
55.1
52.1
38.0
Figure 5: Ablation analysis of SynVRs. Vari-
V
V
66.0
56.2
53.9
39.5
ants exhibit consistent improvements.
67
FlowVerse CoT-E (%)
MathVerse CoT-E (%) 53.9
54
MathVerse CoT-E (%)
53.9
66
52.5
65.632.7
53
54
66
65-
64.8%
64.6,1.4
52
65.0
53
63.9
65-
52.5
64
50.5
51
64
Type w/o Type w/o Type w/o Type
All
DPO
Softmax-DPO
(a) Impact of different error types, where All indicates types are used. Figure Ablation analysis of the proposed Knowledge Internalization Reward.
(b) Impact DPO variants.
improves both CoT-E and final accuracy across Math Verse and Verse. The best results come from combining VSR and VPR, improving CoT-E by up to +3.0% and accuracy by +1.7% on Math- Flow Verse the baseline. These results confirm that VPR ensures geometric precision while VSR stabilizes the global layout, and their combination provides the most reliable perceptual supervision.
Verse, and CoT-E by +2.2% and accuracy by +2. 2.1% 1%
Analysis on Knowledge Internalization Reward. As shown in Figure removing any single Error type consistently degrades performance, indicating that each contributes complementary supervision for internalization. The largest drops arise when excluding omission/misbindiig primitives or con- tradicting geometric constraints, highlighting that correctly binding primitives and respecting core geometric constraints are most critical for keeping reasoning tied to perception. Figure 6 compares different training strategies during training. Vanilla DPO (Rafailov et al., 2023) with all categories offers moderate gains, whereas Softmax-DPP achieves the best results, reaching 66.0% 56.2% on Verweerse. This suggests that fine-grain fine-grained, within-problem preference modeling better captures the nuances of perception-groundee reasoning.
Analysis on Visual-Gated Policy Optimization. As shown Figure 7 the distribution visual reward values shifts steadily upward among different post-traiming methods, with VGPO producing both higher medians and more concentrated high-reward samples. This indicates that VGPO further enhances stability and fidelity by explicitly introducing visual-gated mechanism. To disentangle the effect of VGPO from the effect the gate itself, we conducted an additional ablation of the visual gate, as shown in Figure 8 These results indicate that visual gate is also beneficial during inference, yielding a consistent absolute gain of around 0.6-1% 1% accuracy even when the model is trained without VGPO. Moreover, VGPO provides additional and more substantial improvement, because it uses the visual gate during training to shape the policy itself, rather than only filtering outputs at test time.
Error Type Analysis. As shown in Figure we employ GPT-5 (OpenAl 2025) to classify each re- sponse on the Flow Verse benchmark into one of four categories: Perception Error, Knowledge Inter- nalization Error, Reasoning Error, and Correct. The first row covers specialized visual-math MILMS (i.e., MultiMath-7B (Peng et al., (024a) and SVE-Math-7B Zhang et al., (025b)), an GRPO-style model (i.e., VLM-Rİ (Shen et al. (0255). )) the decoupled method (i.e., MathFlow-7B (Chen et al.) 2025a)), and GPT-4o (OpenAl 2024): the second row traces our variants from the SFT+GRPO baseline through +Syn'V VRs, +lntIRR, and + +VGO to COGFLOW. We observe: (1) GPT-4o's strong overall performance is driven by superior reasoning, yet its diagram perception lags behind special- ized 7B models, indicating that strengthening perception and internalization remains essential; (2)

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Published as a conference paper at ICLR 2026
Vlose CoT-E
MathWerse CoT-E
Reward Value
0.75
67-
53.9
-44
66.0
0.50
66
53
52.5
52.1
0.25
65+
52
64.4
64.5
0.00
50.9
SFT
GRPO
VGPO
63.4
Figure 7: The distribution of visual reward values among different post-training meth- ods. A higher concentration of values indicates stronger perceptual grounding achieved by the corresponding training strategy.
50
None
Inference
Training
Both
Figure 8: Ablation analysis of visual gate. Train- ing and Inference indicates visual gate is only used in training and inference phases respectively.
MultiMath-
SVE Math 7B
GPT-4o
MathFlow- 7B
VLM-RI
21.0%
20.8%
21.7%
35.3%
90.7%
40.6%
41.2%
22 22.8%
20.0%
23.9%
49 7%
19.8%
22.8%
6%
22.1%
14.4%
Baseline
Baseline SVRs
Baseline VAR
Baseline VGPO
COGFLOW
18.0%
0%
16.5%
15.2%
17.0%
48.7%
17.00
54.7%
15.0%
53.5%
17.1%
54.8%
15.0%
59.0%
16 6%
12.3%
11.6%
10 8.8%
Perception Error
Knowledge Internalization Errer
Reasoning Error
Correct
Figure 9: Error-type analysis. We analyze error-type distributions for COGFLOW variants along- side specialized visual-math models, the GRPO-style model, and the decoupled method. The base- line is denoted by the SFT+GRPO setting.
although methods such as MathFlow-7B and VLM-R1 reduce Perception Error and Reasoning Error to some extent, these methods not meaningfully alleviate Knowledge Internalization Error; (3) +SynVRs chiefly reduces Perception Errors (-2%). confirming that geometry -wware and semantic visual rewards improve perceptual fidelity; (4) +IntIzR primarily mitigates reasoning drift, reducing Noowledge-nnteraalizae Error by -2% relative to the baseline; and (5) when all components combined, COGFLOW minimizes every error type while maximizing the proportion of correct pre- dictions. Furthermore, provide systematic case study Appendix $D.13 to analyze the error compensation mechanism.
CONCLUSION
Faithful integration of visual perception into reasoning remains critical yet overlooked challenge in visual mathematical problem solving. In this paper, we observe prevalent reasoning drift issue and propose a cognitive-inspired three-stage framework COcFLOw that explicitly models the hierarchi- cal human reasoning flow from perception to knowledge internalization and finally reasoning. By jointly enhancing perceptual fidelity, enforcing structured knowledge internalization, and anchor- ing multi-step reasoning to visual evidence, COGFLOW effectively mitigates reasoning drift and improves both reasoning accuracy and interpretability. Extensive experiments on commonly used visual mathematical benchmarks validate its consistent advantages over existing MILMMs.
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Published as a conference paper at ICLR 2026
ACKNOWLEDGMENTS
This work was supported in part by the National Natural Science Foundation of China (62402432) and in part by the Fundamental Research Funds for the Central Universities (226-202500080),
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APPENDIX
TABLE OF CONTENTS
General Discussions
17
A.I Ethies Statement
17
A.2 Reproducibility Statement
17
A.3 The Use of Large Language Models
17
Limitations and Future Works
17
Additional Details of MATHCOG Datasel
19
Dataset License and Intended Use
19
B.
2
Data Curation
19
B.3
Dataset Statistics
21
B.4
Dataset Examples
22
Additional Implementation Details
24
Model Initialization
24
The Details of VPR
24
C,3
The Details of IntlRR
24
The Details of InfR
24
C.5 Details of SFT and RL Training
25
C.6 Details of the Reasoning Drift Precision Metric
26
D Additional Analysis
27
D.I Analysis on the Overall Effectiveness of COGFLOW
27
D.2
More Analysis on Synergistic Visual Rewards
"e..............................
27
D.3
The Impact and Efficiency of The Vision Gate
28
D.4
Analysis on Visual-Gated Policy Optimization
................................
29
D.5
Clarifying the Cognitive-Science Analogy
30
D.6
Deeper insights into the mechanism of multi-stage reward structure
30
D.7
Sizes and Architectures Ablation.
31
D.8
Souree of Performance Improvement
31
D.9 Ablations on Individual SynVRs Components
................................
32
DJ 10 Evaluating FG-CLIP for Similarity Reward
32
D. 1l Further Analysis of Error Types
32
D 12 Case Studies
32
D.13 Analysis of Error Compensation Mechanism
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A GENERAL DISCUSSIONS
A.1 ETHICS STATEMENT
Our study focuses on visual mathematical reasoning and does not involve human-subjects exper- iments, personally identifiable information, biometric data. All benchmarks used are publicly available under their original licenses. We introduce МАTICОО (SFT/IntIZRRLL subsets) to sup- port training. MATHCOG contains diagrammatic math items and corresponding solutions (includ- ing synthetic negatives generated by controlled perturbations of reasoning), and-to the best of our knowledge-
no sensitive or personal data. When reusing third-party content, we preserve original attribution, respect redistribution constraints, and release only the metaaataadderivatives per- mitted by the source licenses. We do not identify any foresceable ethical risks associated with this work. We disclose computing resources and carbon-related considerations in the appendix and fol- low common practices to reduce the footprint (mixed precision, batching, and early stopping). The authors declare conflicts of interest beyond those listed the title page.
A.2 REPRODUCIBILITY STATEMENT
To facilitate reproducibility, we have released the full training and evaluation package for COGFLOW, including code, configuration files, and training dataset.
Data, We have released our constructed MATHCOG dataset (including MATHCOG- SFT, MATH- CoG-IntIZR, and MATHCOG-RL. subsets). We have also provided the construction scripts for MATHCoG-IntlRR, including the positive-negative pairing procedure and the five error-type per- turbations used synthesize negative trajectories.
Training and Implementation. We have provided runnable training recipes for the two-stage pipeline (SFT followed by VGPO-based RL), including the base model initialization, optimizer settings, batch sizes, learning rates, training epochs, and hardware requirements. We have released the exact reward formulation and hyperparameters used by VGPO, including the reward weights (ASynVRs; ArntlaR AmmR), the Syn mixing coefficient a, and the visual gate parameters.
Evaluation Protocol. We have released evaluation scripts for all benchmarks used in the paper and report metrics consistent with the main text (answer accuracy and CoT-based evaluation). For baselines whose models code not publicly available, have clearly indicated which numbers are taken from the original papers.
Ablations and Diagnostics. We provide detailed ablation studies $4.3 butions of SynVRs. IntlzR, VGPO, and the visual gate. We further report extended analyses of VGPO and the visual gate in $C.6. All key training hyperparameters and configurations are specified in $ and the visual gate parameters defined in $3.3
covering the contri-
A.3 THE USE OF LARGE LANGUAGE MODELS
large language model (i.e., ChatGPT-5) was used in two ways during the preparation of this paper:
To aid in polishing the writing, including improvements grammar, clarity, and readability. For retrieval and discovery, such as identifying and organizing related prior work.
The model did not contribute to research ideation, experiment design, data analysis, or interpretation of results. Its use was limited non-scientific assistance (e.g., language polishing, rephrasing, and formatting) under direct author supervision. All scientific content, methodological decisions, and conclusions are the sole responsibility of the authors.
A.4 LIMITATIONS AND FUTURE WORKS
While our approach demonstrates strong performance, it has certain limitations. Most notably, the training procedure is computationally demanding, requiring substantial resources to achieve stable improvements. This dependence on large-scale computation may hinder broader adoption, espe- cially in resource-constrained settings.
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For future work, would like to expand our method (especially the method of obtaining visual primitives) to general scenes. While this paper is primarily focused on visual mathematical prob- lems, the procedure for obtaining these primitives is domain-agnostie. The notion of primitives, as a structured representation of visual information, naturally extends to natural scenes beyond the domain of shapes. Concretely, for natural images could first leverage detection segmentation models to obtain instance-level regions and their spatial extents. With the assistance of an LLM and lightweight human verification, these outputs can be normalized into unified primitive schema that encodes structured semantic features for each instance and their relationships. Such a repre- sentation enables a consistent wwwwwlege-intemmaizaaeeee into compact, reasoning-reate state, allowing downstream modules to operate over explicit and verifiable visual evidence.
step that transforms raw perceptual outputs
COGFL.OW Foo's strong performance on visual mathematical reasoning suggests that the pipeline gener- alizes beyond visual math problems. By explicitly decomposing the process into perception, knowl- edge internalization, and reasoning-and training these stages to remain mutually consistent-we obtain a broadly effective framework for visually grounded multimodal reasoning, with the potential to achieve strong performance when transferred to other vision- vision-language domains.
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B ADDITIONAL DETAILS OF MATHCOG DATASET
Existing multimodal reasoning corpora provide abundant natural language annotations but rarely disentangle perception from reasoning (Zhang et al. 2025a Gao al. 2025). This gap makes difficult to supervise models in way that enforces accurate perception and strengthens their ability internalize visual content. To address this, we construct the MATHCOG dataset, which explicitly separates the watching (perception) and rhinking (reasoning) stages. Furthermore, to support the different training phases of COGFLOW, we curate three tailored subsets: MATHCOG-SFT, MATH- COG-IntIZR, and MATHCOG-Ro
B.1 DATASET LICENSE AND INTENDED USE
The MATICоGG dataset will be released under the CC-BY-4.0 license, which permits redistribu- tion and adaptation for both academic research and commercial use, provided that appropriate credit given.
The dataset is intended primarily for research on multimodal reasoning, visual perception, and math- ematical problem solving. It is designed to facilitate studies on perception-reasoningg alignment in multimodal large language models. The dataset should not be used for purposes unrelated to re- search education, including surveillance, profiling, decision decision-making in sensitive domains such as healthcare or law enforcement. We encourage responsible use and proper citation in all derivative works.
B.2
DATA CURATION
B.2.1 DATA COLLECTION
To construct MATHCOG, first collect large pool of geometty-related visual math problems from existing corpora, including MAVIS (Zhang et al. 2025a), Geo170K (Gao et al., 2025). and LLaVA- CoT (Xu et al. 2025a). We then perform careful filtering process obtain a high-quality subset of 111,752 problems. Specifically, we remove (1) problems dominated by weak geometric relevance charts (e.g., histograms, pie charts) or artistic illustrations that deviate from geometric reasoning. and (2) diagrams with either excessively low resolution (< 64) or overly high resolution (> 2048), which would otherwise compromise model training consistency.
MATHCO6-S0 SFT RL Curation
Image
Question
Waatching>
Thinking
<Answer?
figrro. AB
Poler: (6.13, 7,67) O: (:145.5
Using the coredinsio, the midpoii 137
demeter cinle
883
4.63
and Bjo.15.A.53 S3) appeos (6.14.0443)
c. E arde if ZAED 20:; dea 4BCD 2e2 2e 222
Limec 0115, (3.1 -1.90)
Since 2BCD cuesin angle Striingle ACTSS filmns the 2iKD
Labeling
Cireksc 4046, 7.6
Data
Answer 2BCD
Labeled Resporde
MATHCO6 IntizR Curation & Training
Positive Trajectory
Intermalization Inject
Negative Trajectory
Type Omit misbind primitives
Easy
<WATCHING>
888
<WATCHING-
Introduce facts
GPT-5
Manual Check
Type Controdict geometric constroints
AT -THINKING-
Refine
Invoke external theorens inpproppriately
Seftmax-Do
SANSWER:
Type Refer inconsistently to established elements
Figure 10: Curation pipeline of MATHCOG. We label MATHCOG-SS SFT and MATHCOG- RL based on the raw data. To construct MaTHCoG-Intzel we sample positive examples from MATHCOG- SFT and generate five typical negative examples. Then, we adopt Softmax-DPO train the IntlzR model.
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Table 6: Prompt configuration for refining the MATHCOG dataset.
System:
# Your Role: expert math geometry teacher
#t Objective
You will be provided with visual mathematics problem, along with its perception output (<WATCHING>> and a raw solution (<THINK ING>).
Your task to refine the solution such that the reasoning process explicitly internalizes the perceptual information before carrying out logical inference.
## Output Format
The output format should strictly follow the example following:
Question: XXX
<WATCHING>: XXX
<THINK NNG?: XXX
Refined <TRINK NNG?: XXX
Now. here is the data you need to refine:
Question: {question]
<WATCHING>: {watching)
<THINKING> (raw): (solution)
Refined <THINKING>:
B.2.2 MATHCOG-SFT AND MATHCOG-RL CURATION
Visual Primitive Annotation. To construct perception-enhancee dataset, we developed prepro- cessing pipeline as illustrated in Figure 10, Since the raw dataset lacks sufficient perceptual infor- mation, we first curate additional geometric details from the original data. Formally, let denote the raw image. We apply OpenCV-basee operators to extract set of primitive geometric elements:
Extract[I: lev),
where 8.y represents the parameters of the extraction process.
We utilize the OpenCV library to detect sokxellcorddinte-biiee based visual primitives in the images from the collected data, including endpoints and intersections line segments, as well as circular features. For circles, directly extract center coordinates (ar, y) and radius in pixel space. For points (e.g., vertices and character annotations), we similarly obtain precise pixel coordinates and establish optimal correspondences between detected visual elements and character labels via the Hungarian algorithm, thereby obtaining final mappings such as A (æ1, V1) and B (zz, U2).
To unify the representation, the extracted elements are normalized and remapped fixed coordi- nate system, following the normalization scheme in prior work (Wei et al. (2024):
Normalize(E) E') x 20 - 10,
(10)
where normalization is performed by dividing coordinates by the image width and height, and the subsequent linear transformation maps the results into the range [-10, 10).
The complete pipeline can thus be summarized
E- Annotate(Normalin (Normalize((xxtrac(( /; 0..)) x 20 - 10; H).
(11)
This ensures that the final dataset not only captures essential geometric primitives but also provides high-quality, standardized representations suitable for subsequent training and evaluation, thus form- ing the perception part.
Reasoning Annotation. Based on these visual primitives, we further construct reasoning trajecto- ries from the solution rationales in the collected data. The goal is to (i) explicitly internalize the structured primitives into symbolic representation, and (ii) perform step-by-step logical inference over this internalized state to answer the question. These trajectories are first generated with the assistance of GPT-5 and subsequently verified and refined by human annotators. The prompt shown in Table 6 Finally, MATHCOG-RL is created by sampling 10,000 examples from this dataset, while MATHCOG-SFTI is generated by sampling 100,000 examples.
A dedicated team of 31 professional annotators carried out this quality-assurant protocol over a one-month period, ensuring high-fidelity and consistent ground-truth annotations.
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Easy
Medium
Hard
Type 1 Omit misbind primitives
Question:
Ground Truth
In any parallefogram, the perimeter of triangle CDM 5, directly
the segment effactively adds 1 to the total
the figure, the diagonals ABCD intersect point
Type 2 Introduce nonexistent facts
perpendicular line OM is
and AD CD.A A
<WATCHING>
Since A(-4.2, 2.06) and D(9.6, 2.18) differ mainty in x while having same y.
verticaf; should correct the imbalance, so the perimeter is 11.
drawn from point O to AC,
intersocting AD at point the perimeter triangle
Type
Contradict geometric constraints
Since A: (-.222 2.06) B: (-8.32, -3.54) the opposite sides look horizontal and vertical, ABCD is rectangle ..... Estimating AD
CDM is 40 cm, then the
perimeter of purallelogram ABCD
<THINKING>
CD + leads to Perimeter 2 (3.3)=1 =
Type 4 Invoke external theorems inappropriately
Because the diagonals meet of O/0.55, -0.75) we have OC,
GPT-5 Edit
uoo.s Thus AD + CD equals the perimeter of triangle CDM plus 1 f.e., 6, and the peraffelogram perimeter is 2 12.
Type 5 Refer inconsistently to established elements
<ANSWER>
Answer: 10
Because diagonals in paraliologram bisect each other at 0(0.55,- 0.75), ..... the perimeter corresponds to the largest option among moderate choices.
Figure 11: The illustration of data for Knowledge Internalization Reward.
B.2.3 MATHCOG-INTLZR CURATION
Building on MATHCOG-SFT, we construct MATHCOG-InllR as a set of contrastive reasoning trajectories with explicit perturbations:
We first sample 10,000 examples as positive samples from MATHCOG-SED for MATHCOG-Intlak
As mentioned earlier, categorize the errors into five types: (1) omitting misbinding primitives, (2) introducing nonexistent facts, (3) contradicting geometric constraints, (4) invoking external the- orems inappropriately, and (5) referring inconsistently established elements. The negative data for MATHCoG-Inlzl is constructed based on these five error types. The curation of MATHCOG-Iniee IntlzR illustrated the bottom of the Figure I0 We first use LLM, such as GPT-5 (OpenAl 2025). to generate negative trajectories conditioned on each positive trajectory by explicitly instructing it to modify the corresponding "Thinking" part, as specified in the prompt shown in Table 7 The generated candidates are then manually checked. For example, for the Type I error: Omit mis- bind primitives in Figure [1 we input the designed prompt into the LLM and obtain the variant "In any parallelogram, if the perimeter of triangle CDM is 5..... which deliberately alters the rele- vant content in the "Thinking" part of the original positive example. Specifically, for every positive trajectory generate five distinct negative counterparts, resulting in 10,000 positive and 50,000 negative trajectories for Softmax- training.
The easy/medium/hard labels characterize the relative difficulty for the model to detect and correct different types of reasoning drift errors, rather than the difficulty constructing these cases. Easy errors are typically local and isolated (e.g., minor mistake in single visual primitive or reasoning step) and can often be corrected using limited contextual information. Medium errors involve mul- tiple interacting primitives or steps and require the model to integrate information across a broader context. Hard errors are globally entangled with the full reasoning chain, where correcting them de- mands coherent understanding of both the visual configuration and the multi-step logical structure of the solution.
B.3 DATASET STATISTICS
X provides the statistics for the MATHCOG dataset, detailing the number of problems and the distribution of data across different categories. The dataset consists of total of 121,730 problems, with 100,000 labeled for supervised fine-tuning (MATHCOG-SPP
Table
and 10,000 labeled for rein- forcement learning (MATHCOG-RLD) Additionally, there 10,000 positive samples and 50,000 negative samples used in the MATHCOG-Inie (Visual- Auggel-e Reward) subset. validation
set contains 1,730 examples.
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Table Prompt for error type generation MATHCoG-InllR.
Omit Mishind Primitives:
Starting from the correct reasoning. deliberately omit or misbind least one basic geometric primitive (points, lines, circles, etc.), while keeping the overall reasoning fluent and locally plausible. Positive Data: {positive data}
Introduce Nonexistent Facts:
Starting from the correct reasoning, introduce at least one geometric or numerical fact that is not supported by the given figure or problem statement, but make the reasoning appear locally coherent and natural. Poositive Data: {positive data]
Contradict Geometric Constraints:
Starting from the correct reasoning, modify the chain of thought so that at least one step violates the true geometric constraints (e.g., equal lengths, parallelism, angle measures), while preserving a seemingly reasonable narrative.
-Positive Data: {positive data}
Invoke External Theorems Inappropriately:
Starting from the correct reasoning. inuppropriately invoke at least one external theorem formula whose preconditions satisfied, whose is justified by the internalized structure, while keeping the explonation linguistically smooth.
Positive Data: {positive data}
Refer Inconsistently to Established Elements:
Starting from the correct reasoning, alter the chain thought that references previously established elements (points, lines, relationships) become inconsistent, such as swapping labels or changing properties across steps, but without breaking the overall fluency of the text.
Positive Data: {positive data}
Table 8: Statistics of MATHCOG.
Statistic
Number
Total Problem
121,730
Number of MATHCOG-SFT
100,000
Number of MATHCOG-RL
10,000
Number of positive data MATHCoG-IntlAR
10,000
Number of negative data of MATHCOG-Intia
50,000
Number of data for validation
1730
Maximum question length
780
Maximum watching length
1,322
Maximum thinking length
1,288
Average question length
186
Average watching length
463.03
Average thinking length
527.79
Figure shows the length of the various components. The maximum question length is 780, while the maximum lengths for the watching and thinking components are 1,322 and 1,288, respectively. On average, the question length is 186, with the watching and thinking components averaging 463.03 and 527.79, respectively.
B.4 DATASET EXAMPLES
Figure 13 illustrates three representative samples from the MATHCOG dataset (covering both SFT and subsets), explicitly demonstrating our unique data structure that disentangles perception from reasoning. Each sample is organized into four distinct components: the raw visual input (<IMAGE>), the problem statement (<PROBLEM>), the perception trajectory (ettoorrrywwwwt (sWATCHING>), the reasoning trajectory (<THINKING>), and the final answer (<ANSWER>).
Specifically, the perception trajectory translates raw diagrammatic pixels into structured geometric primitives (Points, Lines, and Circles) with precise coordinates in visual parameterized space. For instance, in the leftmost example involving a rectangle, the model identifies specific coordinates for vertices A and B (e.g., A (-7.66,3.5) and the intersection point O. Similarly, in the center
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WATCHING WATCHING THINKING
2,07
2,05
QUESTION QUESTION Avg
2022
Percentage
200 400
620
300
1300
1200
Word Length
Distribution of Word Length among QUESTION, WATCHING, THINKING. We present the distribution of word length, with the horizontal axis representing word length and the vertical axis depicting the corresponding probability distribution.
Figure 12:
-IMAGE>
-IMAGES
<IMAGE>
APROBLENS EM--
<PROBL EM-
<PROBL EMS
ofrectarels
figure, cicle Ce peseeddooaar point
peiat cAOD (quak(()
Sex
length
CW
<WAT CHING-
<WATCHING
Point:
Paint:
Poiat:
Art--5s 35 (-.55
(9.52 -184)
(=1182, Br (-492,
A: (0113 7.67 0: 61146, 4.60)
Liex:
1.--77 (17 1353 3.7.6.35))44 3.52)- 3.53 3.55)
(3,33, 8.14)
6 (5 20)- (7.99
3,50)
Cirde: (2:52 -048, 7.84)
(0.0),
<THINKING>
<THINKING>
1. Rectanglos have propertics vaer dispozab bisect euch obet This more OC and
Since a clamcie and gioon ss maiu OB i-4, The chord OP parpesicular
1. Since AB is dianctes the angle sutradded by my poin the cindle sach C, right ingle. Theretee, 4ACB 90.
point Pand OP is
Theraforn All is
AOB,
the fill chard BC nore tike n tnimos - teree 23a)e(3) - #hapt)3)
movem /FCD
Figure 13: Representative cases in the MATHCOG-SFT and MATHCOG-RL.
and right examples, the model explicitly parameterizes circles by identifying center coordinates and radii, e.g., Circle: (2.52. -0.48, 7.84).
The reasoning trajectory section then performs logical inference anchored in these extracted visual For example, in the rightmost case, the reasoning process explicitly links the perceived fact that "AB is the diameter" (derived from the coordinates <WATCHING> to the geometric theorem that the angle subtended by diameter is 90", leading the correct deduction that LACB 90"5. This structured format ensures that the final <ANSWER> is derived from a cognitive flow of perception cntmmalization>reasonn
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C ADDITIONAL IMPLEMENTATION DETAILS
C.1 MODEL INITIALIZATION
As shown in Table initialize COGF1.OW with 7B-scale pretrained MILM backhone, Qwen2.5-VL-77 (Bai et al. 2025). All parameters of the vision encoder, perception modules, and language decoder updated during training without freezing any component.
C.2 THE DETAILS OF VPR
The Visual Parameterized Reward (VPR) metric quantifies perceptual accuracy within the parameter space of geometric primitives. As depicted in Figure 3 VPR first translates structured visual infor- mation into parametric representations. Critically, such structured inputs constrained three primitive types: points, lines, and circles, which constitute the foundational elements of geometric constructions Trinh line models to the input data; for circles, parameters are extracted directly as center coordinates and radii. Subsequently, an optimal one-to-one correspondence between predicted and ground-trute primitives is established via a multi-class Hungarian algorithm. This matching minimizes the total assignment cost SyPR. The complete computational procedure formalized Algorithm
(2024). For lines, parametric representations derived by fitting straight-
C.3 THE DETAILS OF INTI.ZR TRAINING
As shown in Table T0 the Knowledge Internalization Reward (IntIzR) model is trained on the LATHCoG-IntlZR subset containing 10k pairs (one positive and five negatives per pair). We initial- ize the reward model with Swen2.55VL-33. (Bai al. 2025) using the HuggingFace Transformers library, and train it for 3 epochs with batch size of 64 and learning rate of 7 10-6, it worth noting that we adopt Softmax- DPO for training, with the objective function defined in Eq. 2 where the temperature parameter 3 is fixed to 1.
IntlR is implemented as a reward model trained using Softmax- DPO on contrastive trajectory pairs generated in the MATHCOG-IntlZR subset. For each positive trajectory, use an LLM to generate five negative trajectories by injecting one of the five structured reasoning-error types into the "Think- ing" part, and all synthetic negatives are manually verified. Each trajectory is encoded into hidden representation h, and IntlzR learns a scalar score RImtER (h) E [0, 1] that reflects its consistency with the visual evidence. During training, IntlzR is optimized with the Softmax-DPa objective so that positive trajectories receive higher scores than their corresponding negative trajectories.
C.4 THE DETAILS OF INFR
Accuracy Reward. The accuracy reward evaluates whether the response correct. Following Shao al. (2024), assess correctness solely based the answer field: the reward is I if the answer is correct and otherwise.
Table 9: SFT configurations.
Table 10: IntlzR tions.
configura-
Table 11: RL configurations.
Config
Setting
Config
Setting
Config
Setting
epochs
epochs batch size
batch size
64
epochs
3
base learning rate
10
batch size
64
base learning rate trajectory sampling
x 10 8
optimizer
Adam W Cosine
Base learning rate
7
10
LR scheduler
optimizer
AdamW
rollout generation precision
vLL.M
weight decay
0.01
1
11 oatt enabled
gradient clipping
memory optimization
7eRO-2
gradient accumulation
memory optimization
ZeRO-2
KL penalty coefficient
0.001
memory optimization
ZeRO-2
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Algorithm 1: Multi-Class Matching for VPR
Input:
Ground truth primitives G {G1, C2...., Gm3 with class labels {point, line, circle),
Predicted primitives p= (P, P2... P2.....PPA. P..) with class labels E {point, line, circle}
Output: Optimal matching H Px
Total VPR cost SvPR- (W.)E [ G
H - 0: SyPR f- X {point, line, circle}
// Primitive classes
foreach class & do
Gk + [G; E 9 (aaas(G;)] = k} Pk [P] E class kk)
// GT primitives of class // Predicted primitives of class k
G > and [Px] > then
// Construct cost matrix C₂ E BlaN
for i +
to [9x do
for j + to iPx do
Cx | (Gx (P // L2 distance in parameter
space
end end
// Apply Hungarian algorithm for optimal assignment
Hr + HUNGARIAN(C2)
// Returns matching pairs H4 C x Pa // Class-specific cost Σ Gilla
Sa + E(.DEH Cali,
H HUHr
SvPR + SyPR + Sk
end
else if [h then
SvPR - SvPR + AEN |9k]
// Penalty for missed GT
end
else if [P&s > 0 then
SvPR Syrn App [P&l
Penalty for false positives
end
end return H. SYPR
Format Reward. The format reward verifies whether the response adheres to the required output schema: the model must produce a JSON-style response, i.e. <WATCHING> </WATCHING> <THINKING> ... </THINKING> <ANSWER> </ANSWER>. returns if the response is compliant and 0 otherwise
Finally, we have the Inference Reward RInfRi
RInfR RAce + Re
(12)
C.5 DETAILS OF SFT AND RL TRAINING
For supervised fine-tuning (SFT), we utilize the MATHCOG-SFT subset, which consists of 100,000 curated samples. Both the MATHCOG-SFT and MATHCOG-RL splits are organized into three <THINKING>
<ANSWER>. In our training pipeline, we first perform supervised fine-tuning (SFT) the MATHCOG-SFT. where the model is trained on the "Watching" sequences together with the corresponding <THINKING> and<ANSWER> to strengthen its basic perceptual and reasoning abilities. The model is optimized for I epoch with batch size of 64 and a learning rate x 10 15 We adopt AdamW as the optimizer, using a cosine learning rate scheduler, weight decay of 0.01, and gradient clipping at 1.0 to ensure stable updates.
For RL, the training conducted on the MATHCOG-BR subset (10k samples) for 1 epoch, with a learning rate of 1 x 10-6 and batch size of 16. For each input, sample 8 candidate trajectories
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using lemperatuure-oontollled decoding. The rewards computed by combining Synergistic Visual Rewards (SynVRs), Knowledge Internalization Reward (IntlzR), and Inference Reward (InfR), with their weights fixed to (1, 1, 1). We implement reinforcement learning under the VERL (Sheng al. 2025) framework on NVIDIA A A100 GPUs, with distributed acceleration provided by vLLM. Mixed-precision training is conducted in bfloat 16, 6, and gradient accumulation is employed to simulate larger effective batch sizes, We adopt DeepSpeed ZeRO-2 for memory-efficien optimiza- tion, and the KL penalty coefficient is set 0.001 to stabilize policy updates.
C.6 DETAILS OF THE REASONING DRIFT PRECISION METRIC
As illustrated in Figure ! we assess the precision of reasoning drift: whether each reasoning step is faithfully grounded the perceived structure. Concretely, a generated solution decomposed into steps fs, 1s,).- From each step. extract the referenced visual cues (primitives and relations) R, by GPT-5 OpenAl )2025), and from the perception parse y"* we obtain the set of perceived elements P. A step is deemed grounded if its visual claims are entailed by the perception, formalized by consistency predicate, i.e., (88)) = The precision of reasoning drift detection for a response is
given by
Prec
T Σ
1 (():))
(13)
which measures the fraction of flagged steps that are truly ungrounded; higher values indicate fewer false positives and better alignment between perception and reasoning.
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D ADDITIONAL ANALYSIS
D.1 ANALYSIS ON THE OVERALL EFFECTIVENESS OF COGFLOW
Figure 14 compares the performance of the base model (Qwen2.5-VL-77) the model after SFT, and our full framework (CoGFLOW). The base model achieves 46.30/40.10 10 on CoT-E/Ace, while SFT brings moderate improvements Building upon this, COGFLOW, trained with visual rewards and reinforcement learning, achieves substantial gains of 66.01/56.22, 22, outperforming SFT by +15.31/41..32 (around +30.2%/431. 1.0%% and the base model by 142.6%/440.22 Notably, CoT-E consistently surpasses Acc across all set- tings, with the largest margin observed in COGFLOW (9.79 points), indicating that our approach not only improves final answer accuracy but also significantly enhances the reliability and coherence of intermediate reasoning chains.
50.70/42.90 (+4.40/42.80; approximately 19.5%/47.0%)
Finally, as shown in Table 12 COGFLOW attains overall score of 76.8 (ALL), exceeding recent 7B-class multimodal reasoners and surpassing several general-purpose systems reported here (e.g. GPT-4o 63.8, Claude-3.5-Sonee 67.7, Gemini-2.0-Flash 73.4). While Doubao-pro-1., reaches higher ALL (79.5), COGFLOW delivers the strongest category scores. The pronounced gains on geometry-centric GPS and text-heavy TQA align with our design: SynVRs stabilize fine- fineggrineed perceptual parsing, IntlzR enforces faithful internalization of visual cues, and VGPO optimizes reasoning under multi-signal feedback. Collectively, these components improve not only answer accuracy but also the consistency "see correctly = internalize faithfully = reason coherently" across Math Vista's diverse task types.
D.2 MORE ANALYSIS ON SYNERGISTIC VISUAL REWARDS
We assess perception fidelity by comparing the model's structured output with the ground-truth primitives. Let grod and ge denote the predicted and gold sets of primitives (points/lines/cireles)... An LLM-based matcher sequentially aligns predicted elements to ge (type- and parameter- consistent), yielding a match set M. We compute
[M]
| [ea]
Precision Rocall
Precision = [Epred]
Recall
F1
(14)
Precision + Recall
Figure 15] shows the F1 distribution under three settings (w/o VSR, w/o VPR, w/ VSR): remov- ing either reward lowers perception fidelity, while enabling VSR produces the right-most, highest- centered distribution, indicating more primitives are correctly grounded; VPR provides complemen- tary but weaker regularization.
1.0
$6
Value
0.6
50.70
07 0.4
42.90
<0.10
0.0
Quend 5-V1-78
MFT-Made
CocFrow
w/o VSR
w/o VPR
w/ SynVRs
Figure 14: Effectiveness of COGFLOW. We compare the base model (Qwen2.5-VL. 7B) and its SFT-Model against COGFLOW.
Figure 15: Perception F1 of SynVRs vari- ants. The W/SynVRs achieves the best per- formance.
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Table 12: Accuracy (%)
results on Math Vista dataset.
Method
All
FQA
GPS
MWP
TQA
VQA
Gemini-1.0aal 1.0-rro TTam al.
2025
47.7
48.3
35.1
50.5
65.8
42.5
GPT-4V A 2023
49,9
43.1
50.5
57.5
65.2
38.0
GPT-4o OpenAl
7024)
63.8
Claude 3.5 Sonnet (Anthropic
2024
67.7
Gemini-2.0-Flash (Tem al.
2025
73.4
Doubao-pro-1.5 (Seed et at.
79.5
77.7
88.9
86.0
82.3
62.0
LLaVA-1.5-7B [Liu et al.
2024b)
20.0
22.7
7.7
11.8
26.6
33.0
LLaMA-Adapter-VZZ2
Gao al.
2023
23.9
21.2
25.5
11.3
32.3
31.8
TinyLLaVA (Zhou al.
22244
26.7
20.4
19.7
23.1
44.9
31.8
LLaVA- 1.5- 1.5-13B 13B (Luu ct al.
20246
26.8
19.7
20.2
18.8
45.6
36,9
mPLUG-OWI2-7B (Ye et 2024)
22.2
22.7
23.6
10,2
27.2
27.9
MiniGPT-y2-78 (Chen ct
202.0
23.1
18.6
26.0
13.4
30.4
30.2
G-IJaVA-7B (Gao
25.1
19.1
48.7
3.6
25.0
28.7
VCAR (Jia et all 20224b
33.7
30.9
34.6
38.7
37.3
SPHINX-Plus (Lu el al.
2024a
36.7
54.6
16.4
23.1
41.8
43.0
SVE-Math-7B Zhang el al. 2025b
37.4
31.9
53.9
29.0
41.4
30.8
MultiMath-7B PPng
2024a
50.0
40.1
66.8
61.8
50.0
33.0
X-REASONER (Lm al.
2025
69.0
VL-Rethinker-78 [Wang ct al.
2025a)
73.7
ReVisual-BR R1 (Chen ct al. 2022cc
73.1
WeThink Yang et a..
2025)
70.9
Skywork-KIV-38B (Wang al. 2025c
60.6
COGFLOW-7B
70.4
73.7
Table 13: Ablation analysis of visual rewards and RL strategies. We compare GRPO with PPO and DAPO under consistent reward settings.
Training
Method
Configuration
Flow Verse CoT-E(%)
Flow Acc (%)
Time (h)
w/ SynVRs, w/o IntiZR
56.1
45.8
4.1
DAPO Yu et al. 2025
SynVRs, w/o IntizR
59.7
50.7
4.6
SynVRs, IntlzR
62.8
54.1
4.6
w/o SynVRs, w/o Intl2R
55.9
44.6
3.2
PPO
Yu
2022
SynVRs, w/o IntlzR
58.2
48.3
3.7
w SynVRs, w IntlzR
61.5
53.9
3.7
w/o SynVRs, w/o IntlzR
56.7
47.6
4.2
GRPO Shao
2024
SynVRs, IntleR
61.4
52,9
4.7
w SynVRs, w IntlRR
64.4
55.1
4.8
SynVRs. w/o IntleR w SynVRs, w/o IntleR
57.4
48.7
4.6
VGPO (ours)
63.2
54.7
5.3
w SynVRs, w IntlZR
66.0
56.2
6.1
D.3
THE IMPACT AND EFFICIENCY OF THE VISION GATE
The Impact of the Visual Gate in the Training Phase. To assess how the visual gate improves perceptual quality during VGPO training, we analyze the visual-reward scores produced by its con- ditional re-generation procedure. Figure [6] reports three visual rewards score distributions, each computed over the full training set corresponding to different re-generation under visual gate with the threshold т: (i) the first-attempt scores for all instances; (ii) the post-gating scores after one re-generation round, where only instances with Svis < r are re-generated and use their second- attempt scores while the rest keep their first-attempt scores; and (iii) the post-gating scores after second re- generation round, where instances still below т after the second attempt are re- re-generated again and use their third-ate third-attempt scores, while all others keep the earliest score which they pass the gate. We visualize these distributions together with the threshold and compute the correspond- ing pass rates the fraction of examples whose exceeds at each stage. The results show
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Table 14: Analysis of Inference Schemes. We evaluate COGFLOW and the VLM-R1 baseline under these three settings from the FlowVerse with k 3.
1.00
Reward Value
0.75
0.50
Flow Verse CoT-E Acc
Inference Time
0.25
Model
(h/ / 1000 samples)
0.00
VLM-R1-7B single-pass)
56.17 47.63
2.77
Attempt 2nd Attempt Attempt
VLM-RI-7B (best-of-3 full)
59.45 49.59
5.54
Figure 16: The distribution of visual reward scores during training. At- tempts with scores above the "Thresh- old" are accepted.
CoGFLOW-7B (single-pass)
64.51 55.22
2.72
CoGFLOw-7B (best-of-3 full)
66.04 56.17
5.52
COGFLOW-7B (visual gate)
66.03 56.24
3.06
that the score mass shifts upward and becomes increasingly concentrated above т. indicating that the visual gate acts effective perceptual quality-control mechanism that improves perceptual grounding rather than arbitrarily resampling trajectories.
Meanwhile, as evidenced by Table 13 enabling VGPO with the visual gate incurs only modest overhead of 1.3 training hours, which is acceptable for standard MLLM post-training.
The Impact of the Visual Gate in the Inference Phase. Moreover, we add additional experiments further analyze the impact of the visual gate at inference. Specifically, we compare three inference schemes:
Single-pass:
The model generates a full trajectory (<WATCHING>, <THINKING>, <ANSWER>), which is identical standard MLLM inference.
The model generates k full trajectories and selects the one whose final answer matches the ground truth. We note that this is not a realistic inference strategy, as it relies on access to ground-truth truth answers.
Best-of-k full:
Visual gate (best-of-k perception): CoGFLOW samples l alternative perception trajectories, se- lects the one with the highest visual score, and then performs single reasoning pass conditioned on the selected perception.
As shown in Table [4, to enable fair and controlled comparison, we evaluate COGFLOW and an open-source RI-style baseline (VLM-RI) under all three inference schemes with fixed 3. We additionally report the average time required to process 1,000 examples (computed by selecting 1,000 samples and averaging the end-to-end inference time) for each setting.
The results show that COGFLOW outperforms VLM-RI under both the single magle-pass pass and best-of-3 full settings, indicating that the observed gains are not attributable to more aggressive sampling. Moreover, the visual gate (best-of-3 perception) achieves nearly the same accuracy as best-of-3 full responses while requiring substantially less inference time (2.72 h vs. 3.06 vs. 5.52 h). Overall, these results support that comparison is fair, and that COGFLOW'S visual-gated inference pro- vides a more favorable computation-performamn performance trade-off than naive best-of-k sampling over full responses.
The Impact of l in Visual Gate Finally, ablate the perception sampling number k used by the visual gate at both training time and inference time. Table 15 varies the training-time while keeping the inference-time gate fixed to kt 3, whereas Table 16 varies the inference-time k with the training-time gate fixed to & = 3. Across both ablations, k 3 consistently yields a favor- able performance-computatio trade-off, achieving most of the attainable accuracy gains without incurring excessive additional cost. Accordingly, we adopt k = as the default setting for both stages.
D.4 ANALYSIS ON VISUAL-GATER POLICY OPTIMIZATION
Table B presents an ablation analysis across different reinforcement learning algorithms (DAPO Yo
al. 2025 PPO (Schulman et al. 2017), GRPO (Shao al. 2024), and VGPO) under consis-
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Table 15: Analysis of perception sample num- ber & in the visual gate during training. Inference-time visual gate fixed k
Table 16: Analysis of perception sample num- ber k in the visual gate during inference. Training-tiin time visual gate fixed
3.
Floweerse CoT-E
Flow Verse
Avg Inference Time / 1000 samples)
k
Avg Training Time (h)
k
Acc
CoT-E
Acc
64.40
55.13
4.85
64.51
55.22
2.72
66.03
56.24
6.17
66.03
56.24
3.06
66.60
56.52
7.91
66.36
56.55
3.71
Rcc (K)
ACc
40
Uneag renii Numt
Lafeey
36-
coerica tani Hattiiia ormgg
ciPs
34
66
48
52
56
196)
Figure 17: Scalability and broader applicability of COGFLOW
tent reward configurations based on the MATHCOG. Several clear trends emerge. First, the inclu- sion of visual perceptual reward (VPR) and Knowledge Internalization Reward (IntiRR) consistently improves performance across all methods, confirming their complementary roles in strengthening visual grounding. Second, GRPO outperforms both PPO and DAPO in all settings, highlighting its stability and effectiveness in geometry-orienter reasoning tasks. Most importantly, our proposed VGPO substantially surpasses all baselines, achieving 66.0% accuracy on FlowVerse and Math Verse, respectively. This represents an improvement of over +6 points compared to GRPO with VPR and IntlzR, demonstrating the advantage explicitly disentangling perception and rea- soning optimization. Although VGPO incurs slightly higher training costs, the performance gain justifies the trade-off.
D.5 CLARIFYING THE COGNITIVE-SCIENCE ANALOGY
First, the "watching" stage in COGFLOW is not intended to correspond internalization. Its fune- tion is limited to extracting low-level structured visual information from an image such as points, lines, circles, spatial groupings, and relative configurations. These representations are still purely perceptual: they serve as normalized, noise-reduced description of the raw visual scene, but they do not yet constitute conceptual understanding or task-oriented reasoning.
The subsequent internalization stage is distinct and is designed precisely to bridge perception and reasoning. transforms the structured primitives produced in the watching stage into conceptually meaningful representation, integrating visual evidence with symbolic relations and problem-specife abstractions. Internalization therefore plays the role of aligning perceptual inputs with the evolving reasoning state. For instance: Type (Omit misbind primitives) errors stem from incorrect or incomplete internalization of otherwise correctly perceived primitives. They reflect failures in map- ping the structured perceptual tokens into coherent conceptual mate-henne they lie at the interface between perception and reasoning. Type 4 (Invoke external theorems inappropriately) and Type 5 (Refer inconsistently established elements) arise when the internalized conceptual representation is misinterpreted, misapplied, or inconsistently used within multi-step reasoning. These error types are characteristic of reasoning drift under an inadequately grounded internal state.
D.6 DEEPER INSIGHTS INTO THE MECHANISM OF MULTI- STAGE REWARD STRUCTURE
On the one hand, each reward is designed to target distinct failure mode in visual mathematical reasoning: Firstly, the Synergistic Visual Rewards (SynVRs), which comprise VPR and VSR, are designed to address perceptual failures. Figure 5 presents an ablation of the SynVRs components. The results show that adding either VPR or VSR alone consistently improves performance, and the best results obtained when both are enabled. Figure 15 further reports the perception F1 scores
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for different SynVRs variants, indicating that removing either reward degrades perception fidelity. Finally, the error-type analysis in Figure 9 demonstrates that SynVRs effectively reduce perception- related errors, thereby validating their effectiveness. Taken together, these results show that VPR and VSR provide complementary supervision signals: VPR enforces local geometric fidelity, while VSR encourages global perceptual coherence in terms of overall style and layout, and together they form trustworthy visual cues that serve as robust foundation for effective visual mathematical reasoning, which is consistent with the conclusions reported in prior studies (Guo et al. 2025c Chen et al., 2025a). Secondly, the Knowledge Internalization Reward (IntIzR) is designed to alleviate reasoning drift by bridging the perception and reasoning stages, encouraging the model to produce structured, reasoning-ready outputs (i.e., phowledge-internallee 1989h) that provide a more reliable foundation for subsequent reasoning. The ablation study on IntlaR (Figure (6) shows that removing any single error type consistently degrades performance, indicating that each error type provides complementary supervision. The largest drops occur when excluding the omissio/mmisbidding primitives or contradicting geometric constraints error types, highlighting that correctly binding primitives and respecting core geometric constraints are most eritical for keeping reasoning tied to perception. In addition, the error-type analysis in Figure 2 further demonstrates that IntlzR effectively reduces Knowledge Internalization Errors, which in turn leads improved overall performance. Thirdly, the Inference Reward (InfR) ensures task-level correctness and proper output structure at the final reasoning stage (Shao et al.) 2024).
representations (Ryan & Connell,
On the other hand, this multi-stage reward structure provides principled solution the credit- assignment problem: instead of relying on a single sparse task-level reward, the stage-wise signals supply informative gradients at different points of the cerepiioonineemaaizzttooonesssss The ablation on COGFLOW'S components (Table shows that both SynVRs and IntlzR are essen- tial for achieving strong performance, and that the best results are obtained when all rewards are enabled. In addition, the reasoning drift analysis across three representative pipelines (Figure [b) demonstrates that COGFLow's reasoning trajectories are more stable and better aligned with the visual information, which is consistent with the observed performance gains. Moreover, the error- type analysis in Figure shows that, compared with the Buseline+VGPO setting, COGFLOW sub- stantially reduces Perception Errors, wwwlledg--ineemaliztiioo errors, and Reasoning Errors. This indicates that the three rewards together form a coherent cognitive paradigm: improved perception supports more reliable internalization, and IntlzR further aligns the reasoning trajectory with the visual evidence
pipeline.
D.7 SIZES AND ARCHITECTURES ABLATION.
To demonstrate the scalability and broader applicability of COGFLOW, now explicitly evalu- ate COGFLOW across multiple backbone architectures and parameter scales, rather than only on Qwen2.5-VL.-7B (see Figure 17p. Specifically, we report results for Qwen2.5-VI. (3B and 7B) and Intern (2B, 4B, and 8B). The results show that: (1) COGFL.W consistently outperforms mod- els of comparable size, and even with only 2B 4B parameters achieves competitive performance; (2) the performance gains are maintained or even amplified as model capacity increases; (3) the im- provements hold for both the Qwen and Intern VL. families, indicating that the framework is not tied to specific architecture. These findings provide empirical evidence that COGFLOW is scalable and broadly applicable beyond a single model configuration.
D.8 SOURCE OF PERFORMANCE IMPROVEMENT
Since MATHCOG is explicitly designed to support the training of COGFLOW, the method and the dataset are inherently coupled and cannot be treated as fully independent components. Neverthe- less, for completeness, we conduct controlled analysis to further investigate to what extent the observed performance gains arise from the dataset itself versus from the proposed training frame- work. We conducted an ablation study in the Table [3 where we fix the training data to M MATHCOG and vary only the post-training algorithm. Starting from the same base model, we compare stan- dard PPO, DAPO and our proposed VGPO. The results show that all methods benefit from training on MATHCOG, indicating that VGPO consistently achieves the best performance among all post- training methods under the same data. This suggests that the performance improvements cannot be attributed to the dataset alone, and that the COGFLOW framework and its stage optimization play central role in the observed gain.
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Comparison of pre-trained models on FlowVerse. Results show that using FG-CLIP with a ViT-L-14 backbone yields the best performance.
Table 17:
63-
GaT-E (%)
Accuracy (%
MctaCLIP2
FG-CLIP
ViT-B-16
ViT-L-14
VIT-B-16
ViT-I-114
57
CoT-E Acc
CoT-E
Acc
CoT-E
Acc
CoT-E
Acc
54
0.0
0.2
0.1
0.6
0.8
1.0
62.5
53.9
63.2
54.5
63.3
54.6
63.8
55.3
Figure 18: Effect of varying a on Flow-
Verse performance.
D.9 ABLATIONS ON INDIVIDUAL SYNVRS COMPONENTS
We note that the Figure 5 already included ablations of each Vyn'Rs component. However, we did not previously investigate how different mixing ratios between VPR and VSR affect performance.
We therefore conduct ablation study on the Flow dataset (see Figure 18) by varying the weighting coefficient a while keeping IntlzR and the visual gate enabled. The results show that performance peaks
0.6, indicating that balancing VPR and VSR yields the most effective supervisory signal. Moreover, the fact that a = underperforms a = 1 suggests that parameter- level supervision (VPR) contributes more substantially to perceptual fidelity than semantic similarity alone (VSR).
D.10 EVALUATING FG-CLIP FOR SIMILARITY REWARD
We further evaluate the FG-CLIP similarity module by replacing it with several alternative pre- trained models (e.g.. MetaCLIP2 (Chuang et al. 2025 )) of comparable capacity. Across these vari- ants, the FG-CLIP (ViT-L-144 configuration consistently achieves the best overall performance on Verse, indicating that FG-CLIP provides stronger and more discriminative supervision signal for our VSR component than the alternative encoders
D.11 FURTHER ANALYSIS OF ERROR TYPES
From Figure observe that the Baseline+ setting leads only modest reduction in perception-related errors, but more substantial reduction in reasoning-related errors (see also Fig- ure ireinn in the original submission). We explain this phenomenon follows.
First, we view this as natural consequence of the fact that perception, internalization, and reasoning are independent modules: improvements in perception inevitably propagate to, and influence, the subsequent reasoning process. We conduct a systematic case study to analyze the error com- pensation mechanism. the illustrated example, the perception stage exhibits partial errors: the coordinates of point A are misidentified, which in turn corrupts lines AB and AD. Nevertheless, during the subsequent reasoning stage, the model still produces the correct answer by accurately per- ceiving other visual primitives (e.g., points B and C') and performing precise internalization. This analysis reveals a cognitive processing pattern in which response generation proceeds sequentially through perception internalization reasoning, and suggests that strengthened perception and internalization capabilities can mitigate Reasoning Errors even in the presence of localized percep- tion failures.
In addition, among the Syseline+Snn Baseline+Intl2R, and Baseline+ configurations, Baseline+Syn already yields the largest reduction in Perception Errors, further demonstrating that VRs provide effective improvements to the perceptual component.
D.12 CASE STUDIES
As shown in Figure 19 consider a circle with center O and points A, B, on the circle, where LABO 50%. A typical failure mode is primitive misbinding: one baseline incorrectly treats the given condition as LAOB = 50 and then applies the inscribed-angle theorem to obtain LACB = 25. Another baseline perceives the condition correctly but makes invalid geometric inference by asserting LAOB- = 2LABO, which yields ZACB = 50%. In contrast, with internalized primitives and visual rewards (VPR/IntIzR), our model preserves consistent grounding. Since OAA OB,
32

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Published as a conference paper at ICLR 2026
Question:
step Reasoning
Perception Error
In the figure, points A, B, and are on circle O, and <ABO = 50".
the problem, ZAOB is given as 50% Thus, we treat the 50" as 50"By the
the central angle ZAOB, meaning the AB inscribed angle theorem, the angle subtended by the same are at the circumference half the central angle. Therefore, we say that <ACB V/2 ZAOB x 50 = 25"
What the measure of LACB?
Reasoning Error
A. 40" B. 30" 45" D.50"
Answer: A, 40"
Decoupled Reasoning
The figure shows circle with center O and points A, and C lie on the circle. The given angle is ZAOO 50', which formed at vertex by the segments BA and BO, Based that flawed idea, claim that the central angle ZAOB should be twice the inscribed x ZABO inscribed angle theorem again, conclude that ZACB %/2 ZAOB = 1/2 x 100 = 50%.
Perception Correct
angle: ZAOB
50 100"Then, applying the
Reasoning Error
CosFLow (Ours)
<<WATCHING>POottiic (-6,97, 3.32) nB: (33.6, ACC: (3.24 (o: (0.89, 0.42)aaLime)
Perception Correct
6,97, 3.32) - (-4.28, 0.36) - (-159, 2.6 (556) (3.24, (792)n(-6977 3.32) (-5.62,-00) 00.55 - (-
4.28, -4.21) - (-311,, 7.339) 3.32) (-3.22, 1.94) - (0.54, 0.55) -- (0.89, 0.42) -7.33) -- (-1.31, 33.78 (0.55, 0.24) (0.89, 0.42))((33.16 -7.33 (0.82, .7.7 (3.24, 7 Since OA = OB (radii of the cirde), triangle OAB is
Internali zation
0,38, a. wWW.TC/HNGGaaaagggeeeeeetntt isosceles, making ZOAB ZOBA 50% Therefore, ZAOB can be calculated as 180" 2(50") = 80'. In circle, the angle subtended by an the remaining part of the circumference. Thus, ZACB 1/2 LADB % 80 40". </THINKING><nil BatpI//C/Aatoooo//AAssWWWE A. 40" </ANSWER>',
twice the angle subtended any point
Reasoning Correct
Figure 19: An example of the solution generated by COGFLOW
Question:
CooFLow (Ours)
As shown in the Ligura, and F
are the madpoints of All and AD.
off (0.74, 259ju Limelyn6-7.44, -519) (-5.96
--WATCHINGSPPiis A: (2.3, 5.60eit: -5.19) (8.74, (3933 1.04))of: (-5.8,-
(.588. -1.12)66223,
Perception
respectivelly in the quaditiateral
2.50 2.54))(--674,
(3.90 (,90,1104))ne (...--.....--119) 52524
55322 (606
EE - 6, BC 13, and (D 5,
538) (874, 5444))3393, 104) (631, 217) (337-538)) (874, (544) 588, -112) (-264. S and are posited as the midpolnts AB and aD, respecioely. This geomatric pramise is corroboratte by the provided pertept udl coonlinates, as calculating the midpoints of M-2.3. 5.6. 08-214,55 -5 and DC39% 1.04) 100) yreldsresults proximate to the peroeived points B. 5.8. 1111 and F(0074 2.59). The segment EF
then SCDBC
1.22) - (-074, 2.5), /ttp///ccNN---tttttteeeeooo
Internali
Amswer: 30
midline thus SEF
zation
(rac(1] BDS, follows cakulated length SCD"24 ti02
dingonal $B0
We then andlyze Striingle DBCS SBC
given relttoonshpp S512 1272
Since the equality holds, Sterinngle DBCS proven be right angled tranngl the right angle vertex D. Therefore, 55 (foc (22) ttmes ppont base) ttimes toott (helght) utiliiing the perpedicular logs BD and CD. This ylolds S5 (trringe DBC) - ( Wines 12 Vimes wos, which correponnde de to option B. (/I////NKNNGG>/ee wwwnNinn--/nrrlleeee WERK (ntl jnll we/NASSWER>>
$13*2
Reasoning Correct
Figure 20: Example solution generated by COGFLOW under partially erroneous perception.
40AB is isosceles, LOAB LABO 50 It follows that LAOB 180 -20AAB 80 and by the inscribed-angle theorem LACB LLABOB = 40%, which is correct. This example illustrates that disentangling perception from reasoning and supervising both stages with VPR/IntIZR mitigates primitive misbinding and discourages spurious theorem application.
D.13 ANALYSIS OF ERROR COMPENSATION MECHANISM
As shown in Figure 20 we conduct a systematic case study to analyze the error compensation mechanism. In this instance, perception contains partial errors: the coordinates of point A are misidentified, consequently corrupting lines AB and AD. Crucially, during subsequent reasoning, the model produces correct answers by accurately perceiving other visual primitives (e.g., points B and C') and performing precise knowledge internalization. Our analysis reveals a cognitive paradigm where response generation progresses sequentially through worceppionniieemmliizttoooooooii This demonstrates that enhanced perception and internalization capabilities can mitigate Reasoning Errors even when partial perception failures occur.
33


Reviews:
[{"review_id": "Owglq45E0w", "reviewer": "Reviewer_gjDa", "summary": "This paper proposes a novel cognitive-inspired three-stage framework (perception-internalization-reasoning) to enable faithful integration and proper utilization of visual signals for visual mathmatical reasoning.  The authors proposes synergistic vision reward (SVR) consisting of visual parameterized reward and the visual semantic reward and incoporate the reward in the original GRPO algorithm to devise the VRPO algorithm. The author proposes a new visual anchor reward and curates constrastive training examples to mitigate five common types of unfaithful ultilization of visual signals.", "questions": "See the weaknesses above.", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "+ The paper proposes a new three-step framework for faithful visual mathematical reasoning.\n+ It constructs a new dataset MATHCOG featured with three subset with each subset for a stage respectively.\n+ Experiments on FlowVerse dataset and MathVerse dataset show substaintial improvement compared with baseline methods.", "weaknesses": "+ Some important details are missing. For example, how to construct the contrastive pairs in MATHCOG-VAR dataset? How to incorporate the five common types of reasoning error? How is the visual anchor reward $R_{VAR}$ implemented? Is it a reward in VRPO algorithm or is it a training stage between SFT and VRPO? How iis the correctness and format reward $R_{IR}$ implemented? How to compute the visual parameterized reward in the parameter space if the predicted primitive and the ground-truth primitive is not within the same class? (i.e., the ground truth is an ellipse while the predicted primitive is a circle). \n\n+ As in Eq4, the generation of perception output is repeated $k$ times until the SVR reward is large than a fixed threshold. It is eqivalent to best-of-k sampling if the threshold is not met in early attempts. Therefore, it is doubtful whether the comparison with baseline method is fair or not.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Wtaznr5aUv", "reviewer": "Reviewer_xQ8R", "summary": "The paper argues that the existing vision-language reasoners lack high-quality perception and their reasoning drifts away from their perception in their chain-of-thought. To address these issues, the paper proposes a Cogflow framework that encourages the model to internalize the perceptual primitives before it starts solving the problem. Specifically, the paper proposes visual rewards to ensure that perceptual accuracy in parameter space and layout consistency. This acts as a gate to prevent low-quality generations from unrolling and training rewards for RL training. Further, the paper proposes visual anchored rewards which encourages the model to internalize the perception in its chain of thoughts through synthetically generated negative pairs for softmax-DPO. Finally, they propose visual-gated policy optimization (VGPO) as a global optimization of the model policy. As a result, the model starts performing very well on the FlowVerse and MathVerse datasets. The paper also studies the impact of different design choices in ablation studies.", "questions": "mentioned above", "limitations": "", "rating": 6, "confidence": 4, "soundness": 4, "presentation": 2, "contribution": 3, "strengths": "1. The paper addresses an important problem of the lack of perceptual understanding and reasoning drift in VLMs. \n\n2. The proposed method is quite comprehensive where each component ensures that the model learns to perceive images accurately at the primitive level and the model is encouraged to utilize perception via additional rewards. \n\n3. The experimental results suggest that the method works very well on the FlowVerse, MathVerse, and MathVista dataset. The ablation studies showcase the usefulness of different components on final performance.", "weaknesses": "1. Since the paper has too many moving parts, the paper could be written better. Despite decent experience in VL reasoning, I was finding it hard to keep up with a dense introduction with many jargons. The authors should think about how to portray their story in a more simplified manner.\n\n2. The proposed method to get the primitives for shapes (e.g., Circle) does not seem scalable. How will you get primitives for shapes where the actual dimensions are absent like many natural scenes or even synthetic scenes (like CLEVR dataset)? There is very little information about collecting “watching” data in the main text. Overall, the solution is more tailored towards geometry than general-purpose visual reasoning. It is critical to acknowledge that early in the paper to set the expectations right. \n\n3. The evaluation is somewhat limited. It would be nicer to show performance on more evaluation datasets such as LogicVista, MathVision, We-Math, MMMU-Pro, HallusionBench. Currently, my hunch is that the method will suffer on very complex scenes from LogicVista because there are many images within an image and getting those many primitives might break the model.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "RwcpkJLeIV", "reviewer": "Reviewer_joaQ", "summary": "The paper presents CogFlow, a cognitively inspired framework for visual mathematical reasoning that models the process through three stages: perception, internalization, and reasoning. It introduces stage-specific visual rewards to strengthen the coupling between visual understanding and symbolic reasoning. A new dataset, MATHCOG, is constructed to support training and evaluation. Experiments on FlowVerse, MathVerse, and MathVista show consistent improvements, demonstrating the effectiveness of multi-stage perceptual guidance.", "questions": "The author should include additional experimental results, particularly on other recent benchmarks such as MathVision, We-Math, and DynaMath. Since the performance improvement on MathVista is exceptionally large, results on these datasets are important to verify the generality and robustness of the proposed method.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. The paper presents a conceptually clear and cognitively inspired framework that divides multimodal reasoning into perception, internalization, and reasoning stages, effectively bridging visual understanding and symbolic inference.\n\n2. The paper demonstrates a well-organized methodological design, where the layered reward mechanisms are technically coherent and systematically integrated across different reasoning stages.\n\n3. The paper show consistent improvements on three visual mathematical benchmarks, and contributes a new dataset, MATHCOG, that holds potential value for future research.", "weaknesses": "1. The paper mainly reports results on MathVista, where the gains are substantial, but lacks systematic comparisons with other recent benchmarks such as MathVision, We-Math, and DynaMath. This limits the generality of the claimed performance advantage.\n\n2. The paper evaluates only a single model scale (Qwen2.5-VL-7B) without testing different sizes or architectures, making it difficult to assess the scalability and broader applicability of the proposed framework.\n\n3. The paper provides limited analysis of why the multi-stage reward structure improves reasoning performance. The overall explanation remains empirical, and deeper insights into the mechanism behind these gains would strengthen the work's interpretability.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "6ATLY8TX3N", "reviewer": "Reviewer_Wrhz", "summary": "The authors propose CogFlow, a VLM framework that represents images through geometric primitives before performing reasoning, and introduce MathCog, a new dataset annotated with these primitive-level representations. In CogFlow, visual primitives (e.g., points, lines, circles) are learned to be both geometrically accurate (correct coordinates) and semantically consistent with the original image. During training, primitive predictions with low visual score are regenerated using a thresholding mechanism, termed VGPO. At inference time, only similarity-based supervision is applied since ground-truth primitives are unavailable. To further enhance reasoning accuracy, the authors train a logical error reward that identifies inconsistencies in reasoning steps, which is then used to fine-tune the model to improve consistency.", "questions": "1. In Figure 7, why does the Baseline+SVRs setting show limited reduction in perception errors, with most gains coming from reducing reasoning errors?\n2. In Table 4, do all evaluations use the visual gate during inference? What happens if the model is trained without VGPO but still uses the gate at inference?\nDo all evaluations in Table 4 use visual gate at inference? And what is the impact of training without VGPO but using gate at inference?\n3. In Figure 2, what do the easy/medium/hard categories refer to?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. **Strong results.** Achieves significant performance gains over comparable open-source baselines, showing consistent improvements across multiple benchmarks. Notably, the proposed method narrows the gap with, and in some cases surpasses, the performance of leading closed-source models. \n2. **Clear presentation.** The writing is clear and organized, with helpful figures. Despite the many nuances of the proposed approach, the motivation and details are clearly presented (though some additional details may be missing, see weaknesses).\n3. **Substantive contributions.** Introducing a new reasoning framework around improving VLM perception (CogFlow) and a complementary dataset (MathCog) to support the framework. This method and dataset, which results in strong perfomance, is a valuable addition to the field.", "weaknesses": "1. **Insufficient dataset details and transparency.** The description of the MathCog dataset (in the appendix) lacks important specifics about the data collection and annotation process. It remains unclear exactly what tasks human annotators performed, the scale of human effort involved, and the associated cost or time investment. Additionally, the process for generating negative samples should be more explicitly explained (e.g. prompts used). \n2. **Ambiguity in source of performance improvements.**\nIt is difficult to disentangle the contribution of the CogFlow framework from that of the MathCog dataset. Since CogFlow is trained using MathCog, the performance gains might partially stem from the dataset's scope rather than the model's method. A useful comparison would be to evaluate other methods when trained on MathCog to better understand how much improvement arises from the dataset versus the proposed architecture. While Table 4 (with Table 1/2) gives some sense of this improvement, it is unclear how much improvement other methods would achieve given the same dataset.\n3. **Misalignment with cognitive science framing.**\nWhile the paper draws inspiration from cognitive science concepts, the mapping between these ideas and the proposed components is not entirely convincing. For instance, the \"watching\" tokens more closely resemble internalization (structured encoding of visual input) than perception. Similarly, the Visual Assessment Reward (VAR) appears to operate across multiple stages of cognition, including perception (error type 1) and reasoning (error types 3 and 4), rather than targeting a specific cognitive process. This partial misalignment makes the method more confusing, though the link can still provide good motivation.\n4. **Missing experiments.** Missing comparison with SophiaVL-R1 [1], which may have stronger performance. Missing ablations for individual SVR components (VPR and VSR) (or alpha hyperparameter L248), as well as evaluations of how good FG-CLIP is for similarity reward.\n\n**Suggestions:**\n\n1. Spelling (L26 \"grated\"->\"gated\", L27 \"that produce appear\"->\"that appear\"). Fig 3 \"Visual Similarity Reward\" -> \"Visual Semantic Reward\" and Visual Parameter Reward -> parameterized.\n2. Lots of acronyms (SVR, VAR, VGPO, VPR, VSR) makes method a bit confusing, and some acronyms aren't very descriptive (e.g. synergistic visual reward).\n3. Equation 4 does not follow the algorithm as described. As described (L292), k=2,3 is not evaluated if k=1 passes the threshold, but equation 4 implies 2 and 3 are generated and checked (maximized). \n4. Equation 3 should include preferred trajectory (s+) in the denominator sum, otherwise it's not softmax. Also Softmax-DPO should be better cited L272 [2]\n5. Figure 2 is unclear whether the supervised fine-tuning (SFT) stage uses the \"watching\" and new \"thinking\" annotations, as the yellow/green outline suggests watching/thinking are only used during RL.\n6. Expand one-sentence figure/table captions.\n\n[1] Fan, Kaixuan, et al. \"SophiaVL-R1: Reinforcing MLLMs Reasoning with Thinking Reward.\" arXiv preprint arXiv:2505.17018 (2025).\n\n[2] Chen, Yuxin, et al. \"On softmax direct preference optimization for recommendation.\" Advances in Neural Information Processing Systems 37 (2024): 27463-27489.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)