# MindTopo: Can Foundation Models Reason in Topological Space?

arXiv: https://arxiv.org/abs/2609.11900 · HTML: https://mlsanity.com/paper/2609.11900/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Spatial and embodied reasoning, one of 17 directions in LLM reasoning.
- Builds on 42 of its 97 references that are in the map.
- First posted on arXiv 10 Sep 2026.

## Placed in

- LLM reasoning / [Spatial and embodied reasoning](https://mlsanity.com/lr/spatial-and-embodied-reasoning/)

## Builds on (most cited first, 30 of 42)

- [Qwen3-VL Technical Report](https://mlsanity.com/paper/2511.21631/) (2025)
- [CogVideoX: Text-to-Video Diffusion Models with An Expert Transformer](https://mlsanity.com/paper/2408.06072/) (2024)
- [InternVL3.5: Advancing Open-Source Multimodal Models in Versatility, Reasoning, and Efficiency](https://mlsanity.com/paper/2508.18265/) (2025)
- [Emerging Properties in Unified Multimodal Pretraining](https://mlsanity.com/paper/2505.14683/) (2025)
- [AI2-THOR: An Interactive 3D Environment for Visual AI](https://mlsanity.com/paper/1712.05474/) (2017)
- [RAGEN: Understanding Self-Evolution in LLM Agents via Multi-Turn Reinforcement Learning](https://mlsanity.com/paper/2504.20073/) (2025)
- [Towards World Simulator: Crafting Physical Commonsense-Based Benchmark for Video Generation](https://mlsanity.com/paper/2410.05363/) (2024)
- [PhysBench: Benchmarking and Enhancing Vision-Language Models for Physical World Understanding](https://mlsanity.com/paper/2501.16411/) (2025)
- [OmniSpatial: Towards Comprehensive Spatial Reasoning Benchmark for Vision Language Models](https://mlsanity.com/paper/2506.03135/) (2025)
- [WorldArena: A Unified Benchmark for Evaluating Perception and Functional Utility of Embodied World Models](https://mlsanity.com/paper/2602.08971/) (2026)
- [ThinkMorph: Emergent Properties in Multimodal Interleaved Chain-of-Thought Reasoning](https://mlsanity.com/paper/2510.27492/) (2025)
- [EWMBench: Evaluating Scene, Motion, and Semantic Quality in Embodied World Models](https://mlsanity.com/paper/2505.09694/) (2025)
- [BabyVision: Visual Reasoning Beyond Language](https://mlsanity.com/paper/2601.06521/) (2026)
- [VSP: Assessing the dual challenges of perception and reasoning in spatial planning tasks for VLMs](https://mlsanity.com/paper/2407.01863/) (2024)
- [World-in-World: World Models in a Closed-Loop World](https://mlsanity.com/paper/2510.18135/) (2025)
- [Thinking with Video: Video Generation as a Promising Multimodal Reasoning Paradigm](https://mlsanity.com/paper/2511.04570/) (2025)
- [Core Knowledge Deficits in Multi-Modal Language Models](https://mlsanity.com/paper/2410.10855/) (2024)
- [AlphaMaze: Enhancing Large Language Models' Spatial Intelligence via GRPO](https://mlsanity.com/paper/2502.14669/) (2025)
- [HiDream-O1-Image: A Natively Unified Image Generative Foundation Model with Pixel-level Unified Transformer](https://mlsanity.com/paper/2605.11061/) (2026)
- [Mind the Gap: Benchmarking Spatial Reasoning in Vision-Language Models](https://mlsanity.com/paper/2503.19707/) (2025)
- [ESI-Bench: Towards Embodied Spatial Intelligence that Closes the Perception-Action Loop](https://mlsanity.com/paper/2605.18746/) (2026)
- [KiVA: Kid-inspired Visual Analogies for Testing Large Multimodal Models](https://mlsanity.com/paper/2407.17773/) (2024)
- [RAGEN-2: Reasoning Collapse in Agentic RL](https://mlsanity.com/paper/2604.06268/) (2026)
- [TiViBench: Benchmarking Think-in-Video Reasoning for Video Generative Models](https://mlsanity.com/paper/2511.13704/) (2025)
- [NVIDIA Nemotron Nano V2 VL](https://mlsanity.com/paper/2511.03929/) (2025)
- [Physion-Eval: Evaluating Physical Realism in Generated Video via Human Reasoning](https://mlsanity.com/paper/2603.19607/) (2026)
- [MMGR: Multi-Modal Generative Reasoning Benchmark and Evaluation](https://mlsanity.com/paper/2512.14691/) (2025)
- [PhysicsMind: Sim and Real Mechanics Benchmarking for Physical Reasoning and Prediction in Foundational VLMs and World Models](https://mlsanity.com/paper/2601.16007/) (2026)
- [Vision Language Models See What You Want but not What You See](https://mlsanity.com/paper/2410.00324/) (2024)
- [Spatial-DISE: A Unified Benchmark for Evaluating Spatial Reasoning in Vision-Language Models](https://mlsanity.com/paper/2510.13394/) (2025)

## Abstract (arXiv)

Spatial reasoning depends not only on metric properties such as distance, angle, and shape, but also on topological relations that remain invariant under continuous deformation. Cognitive science identifies these relations as foundational to spatial understanding, yet foundation-model evaluations largely focus on metric or viewpoint-dependent relations. We introduce MindTopo, a benchmark of topological intuition across five properties grounded in cognitive science and formal topology: continuity, separation, order, enclosure, and knots. MindTopo evaluates each property at two cognitive levels. Reasoning asks a model to identify topological relations or infer how they change. Planning instantiates a foundation model as a closed-loop agent whose policy selects environment actions. MindTopo contains 11,030 instances across 13 procedurally generated task types with controllable difficulty. We benchmark 14 MLLMs and study agent configurations augmented with image and video generation, including 3 video generative models in planning settings. Every MLLM performs better on reasoning than on planning, and the best-performing model remains far below observed human performance. On Qwen3-VL-2B-Instruct, supervised fine-tuning and reinforcement learning improve reasoning more than planning. Generated observations retain local cues and reach plausible endpoints, but audited rollouts do not reliably follow environment dynamics or preserve topology across transitions. Our website is at https://mind-topo.github.io/