MindTopo: Can Foundation Models Reason in Topological Space?

Do MLLMs share the topological intuition that humans build?

Key Takeaway: Frontier models can recognize a topological relation in a static scene, but they cannot maintain or operate on it across an action sequence.

arXiv (soon) Code (soon) Dataset (soon) Leaderboard

MindTopo overview: reasoning and planning tasks across five topological properties — continuity, separation, order, enclosure, and knots.

5 Piagetian primitives

13 Task types

11,016 Instances

11 MLLMs evaluated

54.1% Best model vs. 97.4% human

Yunfei Ge^1*, Anbang Liu^1*, Qineng Wang^1*†, Johnalbert Garnica^1*

Jianwen Lyu¹, Zihan Wang¹, Reuben Tan², Jianfeng Gao²

Ruohan Zhang^{1, 3}, Yining Hong³, Jiajun Wu³, Manling Li¹

*Equal contribution, †Project lead

¹Northwestern University, ²Microsoft Research, ³Stanford University

The Benchmark

Topological properties are the spatial properties that survive continuous deformation. A rope can be bent, stretched, or coiled, but whether it is knotted does not change. MindTopo organizes them into five Piagetian primitives and probes each at two cognitive levels — reasoning over a static rendered scene, and planning inside an interactive environment.

Continuity

Is there an unbroken path?

Whether a path or surface forms an unbroken whole — among the earliest spatial concepts a child acquires.

Reasoning: 2D Maze, 3D Maze
Planning: Pipe

Separation

One whole, or distinct parts?

The complement of proximity, and the prerequisite for any reasoning beyond an undifferentiated whole.

Reasoning: IKEA
Planning: One Stroke

Order

Which comes before which?

The sequential arrangement of elements along a path, tracked through a transformation of the scene.

Reasoning: Bead, Origami Point
Planning: Swap

Enclosure

What is inside, what is outside?

The inside/outside relation induced by a closed boundary — the Jordan curve theorem, and the holes in a solid.

Reasoning: Sheep, Hole
Planning: Chat Noir

Knots

Truly knotted, or merely tangled?

The "mother structure" that coordinates every other topological relation: crossings, linking, and knotting.

Reasoning: Knots
Planning: Untangle

The 13 MindTopo tasks arranged by topological property and cognitive level, each with a representative rendered scene, question, or gym environment. — All 13 tasks, with a representative scene and question for each. Interactive environments are labelled **Gym Env**.

MindTopo dataset statistics: 11,016 instances across 13 task types, split 73% reasoning and 27% planning. — 11,016 instances, split 73% reasoning and 27% planning. Each of the eight reasoning tasks contributes roughly 1,000 instances; each of the five planning tasks contributes 600. Every task is procedurally generated with controllable difficulty, so the pipelines can produce arbitrarily large datasets.

Leaderboard

Accuracy (%) on the 13 MindTopo tasks, grouped by the five topological properties. Overall, Reasoning and Planning are unweighted means over the 13, 8 and 5 tasks respectively. Every model recognizes topology far better than it acts on it.

Click a column header to sort · scroll the table sideways to see every task

Proprietary models Open-weight models Random chance / human Reasoning task Planning task

Model	Summary			Continuity			Separation		Order			Enclosure			Knots
Model	Overall↕	Reasoning↕	Planning↕	2D Maze↕	3D Maze↕	Pipe↕	IKEA↕	One Stroke↕	Bead↕	Origami Point↕	Swap↕	Sheep↕	Hole↕	Chat Noir↕	Knots↕	Untangle↕
Gemini 3.1 Flash Lite	36.31	50.84	13.06	21.90	60.30	0.00	42.47	0.00	66.97	36.70	12.80	61.00	55.00	25.60	62.40	26.90
Gemini 3.1 Pro	53.54	60.00	43.20	44.10	75.60	26.56	39.80	29.43	75.90	32.00	71.70	78.60	63.00	58.90	71.00	29.40
GPT-5.4 mini	25.47	35.20	9.90	20.40	30.10	0.00	36.24	0.00	59.86	21.73	15.60	29.25	57.98	26.10	26.00	7.80
GPT-5.5	54.13	57.74	48.33	50.90	58.60	21.67	42.67	25.00	72.37	31.75	100.00	78.21	69.26	61.70	58.20	33.30
InternVL-3.5-241b-a28b	21.78	32.74	4.26	10.80	34.50	1.20	17.80	0.00	57.30	25.40	12.80	28.30	50.00	6.70	37.80	0.60
Nemotron Nano 12B v2 VL	21.90	31.63	6.34	18.30	25.20	0.50	41.00	0.60	41.50	24.50	13.90	18.00	59.70	1.70	24.80	15.00
Gemma-4-31b-it	25.50	39.74	2.71	26.50	55.80	7.50	18.80	0.00	66.37	45.24	0.00	50.35	49.90	0.00	5.00	6.06
Qwen3.5-VL-397B	34.94	44.34	19.90	18.72	67.77	0.00	39.76	0.00	62.19	23.21	59.50	42.99	54.49	29.70	45.60	10.30
Cosmos-Reason2	23.18	30.74	11.10	30.10	22.40	0.00	35.74	2.20	56.96	9.82	19.40	29.05	41.82	12.20	20.00	21.70
BAGEL-7B	17.66	27.80	1.44	7.30	33.30	0.00	21.18	0.00	17.42	13.69	0.00	34.73	57.98	0.00	36.80	7.20
ThinkMorph-7B	14.83	24.10	0.00	8.20	31.10	0.00	21.18	0.00	17.22	11.81	0.00	20.00	55.89	0.00	27.40	0.00
Random Chance	9.49	13.27	3.45	11.12	22.83	0.00	19.94	0.00	8.43	10.07	8.89	10.31	8.41	1.67	15.01	6.67
Human	97.40	95.77	100.00	97.18	98.62	100.00	94.37	100.00	95.77	91.94	100.00	99.06	97.65	100.00	91.55	100.00

Highlighted cells mark the best evaluated model in that column. Random Chance and Human are reference rows and are excluded from that comparison.

Citation

If you find our work useful in your research, please cite:

@misc{ge2026mindtopo,
      title={MindTopo: Can Foundation Models Reason in Topological Space?},
      author={Yunfei Ge and Anbang Liu and Qineng Wang and Johnalbert Garnica and Jianwen Lyu and Zihan Wang and Reuben Tan and Jianfeng Gao and Ruohan Zhang and Yining Hong and Jiajun Wu and Manling Li},
      year={2026},
      url={https://mind-topo.github.io/},
}