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REVIEW 4 major objections 5 minor 16 references

The paper argues that vision-language models fail at visually grounded equation solving because counting repeated object instances is the primary bottleneck, even when object recognition is accurate.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-08-04 19:49 UTC pith:GEHUTJQY

load-bearing objection Useful diagnostic benchmark with a solid main result, but the bottleneck attribution is not as clean as the abstract implies. the 4 major comments →

arxiv 2509.09013 v1 pith:GEHUTJQY submitted 2025-09-10 cs.CL cs.AIcs.CV

Can Vision-Language Models Solve Visual Math Equations?

Monjoy Narayan Choudhury , Junling Wang , Yifan Hou , Mrinmaya Sachan This is my paper
classification cs.CL cs.AIcs.CV
keywords vision-language modelsvisual equation solvingobject countingsymbolic reasoningmultimodal reasoningcompositional reasoninglinear equationsbenchmark
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper tries to show that vision-language models fail at visual equation solving not because they cannot read symbols or recognize objects, but because they cannot reliably count repeated object instances. On symbolic equations rendered as text, all tested models achieve above 97 percent accuracy; on fully visual equations, where coefficients must be inferred by counting icons, accuracy falls below 12 percent. A controlled decomposition into recognition, counting, and composition shows recognition is accurate while counting collapses from 74 percent at 2-5 icons to 9 percent at 16-20 icons. The paper also finds that composing recognition with reasoning adds further errors, and that symbolic reasoning itself degrades when systems grow to three variables. If right, current VLMs lack a prerequisite for grounded symbolic computation: reliable visual counting.

Core claim

The central claim is that coefficient counting is the primary bottleneck in visual equation solving, even when variables are recognized accurately. The evidence comes from a hybrid visual-symbolic condition, which removes counting while preserving icon recognition and symbolic reasoning: accuracy there reaches 64.45 percent, far above the fully visual condition but still below symbolic. Direct counting tests show all models struggle, and accuracy drops sharply as the number of icons increases, with a strong negative correlation between count size and accuracy. Composition—recognition plus reasoning—accounts for a secondary, independent error source, and increased equation complexity shifts p

What carries the argument

The central mechanism is the visual equation itself: a system of linear equations in which each variable is an object icon and each coefficient is expressed by repeating that icon. The paper compares four formats—fully symbolic text, visual-symbolic (icon variable, numeral coefficient), fully visual, and isolated counting and recognition tasks—to decompose the task. What carries the argument is the controlled contrast between the visual-symbolic condition and the fully visual condition, which isolates the act of counting from recognition and reasoning.

Load-bearing premise

The conclusion that counting is the primary bottleneck assumes that the visual-symbolic condition—icons with numerals—removes counting cleanly and leaves recognition and reasoning as the only demands; if the hybrid format is intrinsically harder because of layout, icon ambiguity, or extra parsing, the ranking of bottlenecks weakens.

What would settle it

Compare visual-symbolic accuracy (numerals plus icons, currently 64%) with a control where each coefficient numeral is placed beside exactly one icon, so no counting is needed but layout is the same. If this control also stays well below symbolic accuracy (97%), the bottleneck is not counting but something about embedding icons in equations; if it rises to near 97%, counting is confirmed as the primary cause.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

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If this is right

  • If counting is the bottleneck, improving a VLM's visual counting ability should directly raise accuracy on visual equation solving.
  • Because recognition is already accurate, object recognition is not where training effort should be spent first.
  • Composition of recognition and reasoning introduces errors beyond either subskill, so evaluation should measure composed tasks, not subskills alone.
  • As equation complexity grows, symbolic reasoning becomes a limiting factor even in textual form, so math generalization cannot be assumed.
  • A benchmark of this kind can serve as a diagnostic for grounded multi-step reasoning in vision-language models.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • A testable extension: hold icon layout identical across conditions and replace icons with labeled colored shapes to control recognition difficulty; the paper's own data suggest the visual-symbolic gap from 64% to 97% includes a parsing component beyond counting.
  • The counting accuracy decline from 74% at 2-5 icons to 9% at 16-20 icons suggests a systematic failure mode; probing whether models underestimate, merge nearby icons, or lose track after a threshold could reveal the mechanism.
  • The compositional error source implies that training on counting and equation solving separately may not transfer; fine-tuning a VLM jointly on counting plus solving and measuring whether the 64% visual-symbolic gap closes would test this directly.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper introduces a synthetic visual equation-solving benchmark in which systems of linear equations are rendered with object icons as variables and repeated icons (or numerals) as coefficients. Six VLMs (Qwen family, GPT-4o, Gemini 2.0 Flash) are evaluated under direct and chain-of-thought prompting across four conditions: fully visual equations, symbolic/text equations, visual-symbolic (icon variables, numeric coefficients) equations, and isolated counting/recognition subtasks. The main empirical claims are that symbolic equation accuracy is near-perfect (>97%), fully visual equation accuracy is below 12%, visual-symbolic accuracy is about 64%, and direct counting accuracy falls from 74% (2–5 icons) to 9% (16–20 icons). From this the authors conclude that coefficient counting is the primary bottleneck, that composing recognition with reasoning adds further errors, and that three-variable systems expose limits in symbolic reasoning. Data and code are promised as public.

Significance. If the central attribution holds, the paper provides a useful diagnostic for a real gap in current VLMs: visually grounded symbolic computation fails even when textual arithmetic and icon recognition individually succeed. The decomposition into counting, recognition, and composition is a sensible experimental design, and the released synthetic benchmark could serve as a controlled testbed for future multimodal reasoning models. The direct counting results, the error-cascade case study in Fig. 14, and the extension to reasoning models in Table 2 are concrete and valuable. However, the significance depends on whether the visual-symbolic condition truly isolates counting from other visual-parsing factors; that premise is not fully established, and the inconsistency in Table 1's totals weakens confidence in the quantitative claims.

major comments (4)
  1. [§3.2.1, Fig. 2, Fig. 6, Fig. 8] The central claim that counting is the primary bottleneck rests on the assertion that the visual-symbolic condition 'removes the need for counting while preserving the need for icon recognition and symbolic reasoning.' But the fully visual condition contains many repeated icons, increasing object density, overlap, and grouping ambiguity, whereas the visual-symbolic condition has only one icon per term. The 52-point gap between visual-symbolic (64.45%) and fully visual (<12%) could therefore reflect visual clutter/parsing difficulty rather than the arithmetic act of enumerating objects. The isolated counting task (Fig. 8) shows low accuracy, but it is a separate, simpler task and does not demonstrate that counting is the limiting factor when counting must be composed with equation solving. A controlled manipulation that holds the visual input fixed and provides counts externally (e.g., nu
  2. [Table 1 / §2.1] The manuscript states in §2.1 that 'Each experiment is conducted on a set of 1,000 constructed examples and run once per model-setting.' Table 1, however, reports 1,476 + 4,452 + 4,835 + 1,235 = 11,998 total examples across counting result ranges. This is unexplained and inconsistent with the stated 1,000-example dataset. The percentages in Table 1 and the Pearson correlation of –0.90 depend on these denominators. The authors should state the actual number of counting trials per model, explain how the 11,998 total arises, and report per-range sample sizes consistently.
  3. [§2.1, all result figures] Every experiment is run once per model-setting, with no error bars, confidence intervals, repeated seeds, or significance tests. For a benchmark that makes quantitative bottleneck claims, point estimates from a single run are fragile; for example, the visual-symbolic 'overall accuracy' of 64.45% is a single draw. The paper should either report multiple runs with variance, or at minimum provide binomial confidence intervals for the key percentages and a significance test for the counting-accuracy decline across ranges.
  4. [§3.1.1 vs. B.2, Table 2] Section 3.1.1 states that 'all evaluated models ... consistently fail to solve equations in visual form (overall accuracy < 12%)', but Appendix B.2 reports that o4-mini achieves 36.5% and Gemini 2.5 Pro achieves 43.1% on visual equation solving. If these reasoning models are meant to be included in the general claim, the <12% statement is false; if they are considered a separate class, the scope should be stated explicitly and the abstract/main-text 'VLMs fail' framing should be softened to avoid overgeneralization. This discrepancy affects the paper's headline conclusion.
minor comments (5)
  1. [Fig. 7 / §3.2.2] The caption of Fig. 7 says 'Accuracy on variable coefficient counting', but the section describes variable recognition and the y-axis labels are 'Symbolic Recognition Visual-Symbolic'. This mismatch between caption, axis labels, and task description is confusing and should be corrected.
  2. [Fig. 4 caption] The caption says 'all models could solve the equations perfectly', but the text reports accuracy >97% with CoT prompting, not 100%. 'Near-perfectly' would be accurate; 'perfectly' is an overstatement.
  3. [§3.2.1, Fig. 6] The term 'overall accuracy' is used without defining the aggregation across the six models. Specify whether this is an unweighted mean of model accuracies, a pooled accuracy, or a per-condition average; otherwise the headline 64.45% is not reproducible.
  4. [A.4, Recognition Prompting] The recognition-prompting subsection says the example is shown in Fig. 12, but Fig. 12 is the counting prompt; the recognition prompt appears in Fig. 13. The cross-reference should be fixed.
  5. [§2.2 / A.3] The model family is called 'QwenVL-2.5' but the models are referred to as Qwen-3B, Qwen-7B, etc., and the cited reference is the Qwen technical report rather than a Qwen2.5-VL model card. Please align the model naming and provide the appropriate citation for the exact model versions used.

Circularity Check

0 steps flagged

No significant circularity: this is an empirical benchmark with direct measurements, not a derivation that reduces to its inputs.

full rationale

The paper is an empirical evaluation study, not a formal derivation. Accuracy numbers (symbolic >97%, visual <12%, visual-symbolic 64.45%, counting 74%→9%) are direct measurements on generated data, reported as observations rather than as predictions derived from fitted parameters. No quantity is fitted to one subset and then reported as a prediction on a closely related subset. The central claim that counting is the primary bottleneck is supported by controlled comparisons (symbolic vs. visual-symbolic vs. visual; direct counting task; correlation of counting accuracy with icon count). The visual-symbolic condition in §3.2.1 is intended to remove counting while preserving recognition and reasoning; whether it fully isolates counting is a question of experimental validity, not circularity, because the conclusion does not assume the conclusion—it is inferred from the measured gap and from the separate counting task. Citations to IconQA, MathVista, and model papers are external sources and are not used to justify the paper's own empirical claims. No self-citation chain supplies the load-bearing premise, and no ansatz is smuggled in via citation. Therefore the derivation chain, such as it is, is self-contained and the circularity score is 0.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

No free parameters are fitted and no new entities are postulated. The central claims rest on evaluation assumptions about scoring, task isolation, and run representativeness, listed above.

axioms (4)
  • domain assumption Exact-match scoring is a valid measure of equation-solving ability.
    The entire evaluation reduces model output to exact variable-value matches; no partial credit or error taxonomy is used, so claims about solving depend on this scoring choice.
  • domain assumption Text-in-image versus icon-image comparison isolates visual grounding with symbolic content held fixed.
    Section 3.1 compares visual and symbolic formats to attribute failure to visual interpretation; physical differences such as icon density and layout are not controlled.
  • domain assumption The visual-symbolic condition removes counting while preserving recognition and symbolic reasoning.
    Section 3.2.1 states this setting removes the need for counting while preserving icon recognition and symbolic reasoning; this is the load-bearing decomposition assumption.
  • domain assumption One evaluation run per model-setting configuration is representative of model performance.
    Section 2.1 says each experiment runs once per model-setting, so stability across decoding randomness is assumed without evidence.

pith-pipeline@v1.3.0-alltime-deepseek · 8641 in / 13189 out tokens · 150209 ms · 2026-08-04T19:49:30.139012+00:00 · methodology

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Cite this review

Pith. "Pith review of Can Vision-Language Models Solve Visual Math Equations?." pith.science (2026). https://pith.science/paper/GEHUTJQY

@misc{pith2026250909013,
  author       = {Pith},
  title        = {Pith review of: Can Vision-Language Models Solve Visual Math Equations?},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GEHUTJQY}},
  note         = {Machine review of arXiv:2509.09013}
}
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read the original abstract

Despite strong performance in visual understanding and language-based reasoning, Vision-Language Models (VLMs) struggle with tasks requiring integrated perception and symbolic computation. We study this limitation through visual equation solving, where mathematical equations are embedded in images, variables are represented by object icons, and coefficients must be inferred by counting. While VLMs perform well on textual equations, they fail on visually grounded counterparts. To understand this gap, we decompose the task into coefficient counting and variable recognition, and find that counting is the primary bottleneck, even when recognition is accurate. We also observe that composing recognition and reasoning introduces additional errors, highlighting challenges in multi-step visual reasoning. Finally, as equation complexity increases, symbolic reasoning itself becomes a limiting factor. These findings reveal key weaknesses in current VLMs and point toward future improvements in visually grounded mathematical reasoning.

Figures

Figures reproduced from arXiv: 2509.09013 by Junling Wang, Monjoy Narayan Choudhury, Mrinmaya Sachan, Yifan Hou.

Figure 1
Figure 1. Figure 1: An example of our generated visual equations (i.e., [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 4
Figure 4. Figure 4: Performance of VLMs on symbolic equation solv [PITH_FULL_IMAGE:figures/full_fig_p003_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: An example of our generated visual-symbolic equa [PITH_FULL_IMAGE:figures/full_fig_p003_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Performance of VLMs on visual-symbolic equation [PITH_FULL_IMAGE:figures/full_fig_p003_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Accuracy on variable coefficient counting. Results [PITH_FULL_IMAGE:figures/full_fig_p004_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Performance of VLMs on variable counting. Results [PITH_FULL_IMAGE:figures/full_fig_p004_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Overall accuracy across 6 models on solving equa [PITH_FULL_IMAGE:figures/full_fig_p005_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Direct Prompting Template and Example. The same prompt is used across all models for consistency. Two-Step CoT Prompting. To encourage deeper reasoning while avoiding overly rigid output structures, we adopt a two-step chain-of-thought (CoT) prompting strategy. In the first turn, the model is prompted to freely analyze and solve the problem in its own words. In the second turn, we provide both the origina… view at source ↗
Figure 11
Figure 11. Figure 11: CoT Prompting Strategy. The left box initiates free-form reasoning, while the right box extracts the final answers based on the initial prompt and generated response. Step 1: Analysis Prompt Look at this image and identify the count of each object. Provide your analysis step by step and ensure all details are clear. Only use the following allowed objects: apple, palm_tree, strawberry, egg, clover, donut, … view at source ↗
Figure 12
Figure 12. Figure 12: Two-step prompting strategy for solving visual object counting task. [PITH_FULL_IMAGE:figures/full_fig_p008_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Two-step prompting strategy for the object-type recognition task. [PITH_FULL_IMAGE:figures/full_fig_p008_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Response of Gemini 2.0 Flash for the given query image (top) using the CoT prompting. Note the mistake made by [PITH_FULL_IMAGE:figures/full_fig_p009_14.png] view at source ↗

discussion (0)

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Reference graph

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