CurveBench: A Benchmark for Exact Topological Reasoning over Nested Jordan Curves

We introduce CurveBench, a benchmark for hierarchical topological reasoning from visual input. CurveBench consists of $\text{756 images}$ of pairwise non-intersecting Jordan curves across easy, polygonal, topographic-inspired, maze-like, and dense counting configurations. Each image is annotated with a rooted tree encoding the containment relations between planar regions. We formulate the task as structured prediction: given an image, a model must recover the full rooted containment tree induced by the curves. Despite the visual simplicity of the task, the strongest evaluated model, $\text{Gemini-3.1-Pro-Preview}$, achieves only \textbf{71.1\\%} tree-generation accuracy on CurveBench-Easy and \textbf{19.1\\%} on CurveBench-Hard. We further demonstrate benchmark utility through RLVR-style fine-tuning of open-weight vision-language models. Our trained $\text{Qwen3-VL-8B}$ model improves over $\text{Qwen3-VL-8B-Thinking}$ from \textbf{2.8\\%} to \textbf{33.3\\%} tree-generation accuracy on CurveBench-Easy, exceeding $\text{GPT-5.4}$ and $\text{Claude-Opus-4.5}$ under our evaluation protocol. The remaining gap, especially on CurveBench-Hard, shows that exact topology-aware visual reasoning remains far from solved.