Origami as a Spatial and Geometric Reasoning Benchmark

Multimodal language models perform well on visual recognition and on tool use, but these abilities do not transfer directly to spatial and geometric reasoning. Geometric structure does not reduce to text or to a single image: exact angles and incidences, which surface lies in front of which, and the order in which operations were applied are not recoverable from a semantic description of a scene. We introduce CP2Seq, a benchmark for evaluating geometric reasoning and sequential planning in multimodal language models through origami. The benchmark contains 600 procedurally generated samples, each requiring a model to produce a valid sequence of folds from a crease pattern and a target folded state. Samples are produced by a deterministic folding engine. Ground truth is recorded during generation. We release the generation and evaluation framework to support additional samples and experiments across reference depths and action models. During evaluation, the engine executes proposed folds or rejects invalid actions with explicit feedback, while the model remains responsible for selecting actions and searching for a solution. We evaluate multimodal language models under progressively greater tool assistance, and against a deterministic breadth-first search over the same action set. Candidate sequences are assessed by executing them and comparing the resulting folded state with the target, allowing for planar translations, rotations, and reflections. Performance falls off sharply with sequence depth: in the Luna ablations no configuration solves a medium or hard sample (11 or more folds), while some-layers solves rise from 6/30 to 18/30 with higher reasoning effort. Failures are dominated by search rather than by illegal moves: in the broader evaluation, 48% of episodes end by revisiting earlier states. A breadth-first search over the same actions, under its own time budget, solves 54.7% of easy-group attempts, so short sequences are within reach of systematic search.