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 $\textbf{756 images}$ 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, $\texttt{Gemini-3.1-Pro-Preview}$, 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 $\texttt{Qwen3-VL-8B}$ model improves over $\texttt{Qwen3-VL-8B-Thinking}$ from $\textbf{2.8\\%}$ to $\textbf{33.3\\%}$ tree-generation accuracy on CurveBench-Easy, exceeding $\texttt{GPT-5.4}$ and $\texttt{Claude-Opus-4.5}$ under our evaluation protocol. The remaining gap, especially on CurveBench-Hard, shows that exact topology-aware visual reasoning remains far from solved.