The Gordian Knot for VLMs: Diagrammatic Knot Reasoning as a Hard Benchmark

Jigsaw puzzle solving requires jointly reasoning about visual content and geometric constraints, yet existing benchmarks use rectangular cuts that create ambiguous ground truth in texture-repeated regions. We introduce \textit{\ours{}}, a benchmark with tab-and-blank interlocking pieces where geometric constraints provide strong local compatibility requirements that, combined with visual content, yield unambiguous ground truth. Across 95K instances at four grid densities (4$\times$4 to 16$\times$16), we find that \textbf{zero-shot VLMs largely lack geometric reasoning}: only one of five frontier models (GPT-5.5) exceeds random baseline on 4$\times$4 puzzles, while all others perform at chance level. While supervised fine-tuning achieves $>$97% on 4$\times$4, \textbf{all models collapse on larger grids}: GPT-5.5 drops from 70% to near-random on 8$\times$8, and even fine-tuned models fall below 5% on 12$\times$12. This ``scaling cliff'' suggests current architectures cannot maintain consistent constraint satisfaction as the number of pieces increases. \ours{} establishes scalable geometric reasoning as an open challenge for vision-language models.

Vision-language models (VLMs) have made rapid progress in visual perception and increasingly support real-world tasks that depend on images. Many such tasks, however, require more than rec- ognizing what an image contains: a model must use visual evidence to make a complete decision whose parts jointly satisfy global constraints. We introduce COMPLEXITYWORLD, a benchmark of 390 tasks across 39 domain-inspired visual worlds and 29 decision categories. Each task is generated from a hidden structured specification, rendered as a visual scene, and scored by an exe- cutable verifier that accepts any feasible solution. Under direct inference, all evaluated models ex- cept GPT-5.6-Sol remain below 40% verifier ac- ceptance rate (VAR), while GPT-5.6-Sol reaches 75.6%. Performance improves substantially when the same decision information is made explicit in structured form, yet varies sharply across equiva- lent visual presentations. Agent scaffolds provide smaller, model-dependent gains. Together, these results reveal a persistent visual-to-decision bot- tleneck that additional inference alone does not remove.

Reasoning in vision-language models (VLMs) has recently attracted significant attention due to its broad applicability across diverse downstream tasks. However, it remains unclear whether the superior performance of VLMs stems from genuine vision-grounded reasoning or relies predominantly on the reasoning capabilities of their textual backbones. To systematically measure this, we introduce CrossMath, a novel multimodal reasoning benchmark designed for controlled cross-modal comparisons. Specifically, we construct each problem in text-only, image-only, and image+text formats guaranteeing identical task-relevant information, verified by human annotators. This rigorous alignment effectively isolates modality-specific reasoning differences while eliminating confounding factors such as information mismatch. Extensive evaluation of state-of-the-art VLMs reveals a consistent phenomenon: a substantial performance gap between textual and visual reasoning. Notably, VLMs excel with text-only inputs, whereas incorporating visual data (image+text) frequently degrades performance compared to the text-only baseline. These findings indicate that current VLMs conduct reasoning primarily in the textual space, with limited genuine reliance on visual evidence. To mitigate this limitation, we curate a CrossMath training set for VLM fine-tuning. Empirical evaluations demonstrate that fine-tuning on this training set significantly boosts reasoning performance across all individual and joint modalities, while yielding robust gains on two general visual reasoning tasks. Source code is available at https://github.com/xuyige/CrossMath.