# Unfold The World: Factorize 4D Properties in Reinforcing Spatial Reasoning

arXiv: https://arxiv.org/abs/2609.03729 · HTML: https://mlsanity.com/paper/2609.03729/

## 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 63 references that are in the map.
- First posted on arXiv 3 Sep 2026.

## Placed in

- LLM reasoning / [Spatial and embodied reasoning](https://mlsanity.com/lr/spatial-and-embodied-reasoning/)
- Multimodal LLMs / vision-language models / [Embodied and 3D vision-language models](https://mlsanity.com/mllm/embodied-and-3d-vision-language-models/)

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

- [GPT-4 Technical Report](https://mlsanity.com/paper/2303.08774/) (2023)
- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning](https://mlsanity.com/paper/2501.12948/) (2025)
- [Qwen2.5-VL Technical Report](https://mlsanity.com/paper/2502.13923/) (2025)
- [Qwen2-VL: Enhancing Vision-Language Model's Perception of the World at Any Resolution](https://mlsanity.com/paper/2409.12191/) (2024)
- [Gemini 2.5: Pushing the Frontier with Advanced Reasoning, Multimodality, Long Context, and Next Generation Agentic Capabilities](https://mlsanity.com/paper/2507.06261/) (2025)
- [DAPO: An Open-Source LLM Reinforcement Learning System at Scale](https://mlsanity.com/paper/2503.14476/) (2025)
- [Qwen3-VL Technical Report](https://mlsanity.com/paper/2511.21631/) (2025)
- [InternVL3: Exploring Advanced Training and Test-Time Recipes for Open-Source Multimodal Models](https://mlsanity.com/paper/2504.10479/) (2025)
- [Decoupled Weight Decay Regularization](https://mlsanity.com/paper/1711.05101/) (2017)
- [Expanding Performance Boundaries of Open-Source Multimodal Models with Model, Data, and Test-Time Scaling](https://mlsanity.com/paper/2412.05271/) (2024)
- [Matterport3D: Learning from RGB-D Data in Indoor Environments](https://mlsanity.com/paper/1709.06158/) (2017)
- [Are We on the Right Way for Evaluating Large Vision-Language Models?](https://mlsanity.com/paper/2403.20330/) (2024)
- [VLM-R1: A Stable and Generalizable R1-style Large Vision-Language Model](https://mlsanity.com/paper/2504.07615/) (2025)
- [ARKitScenes: A Diverse Real-World Dataset For 3D Indoor Scene Understanding Using Mobile RGB-D Data](https://mlsanity.com/paper/2111.08897/) (2021)
- [VL-Rethinker: Incentivizing Self-Reflection of Vision-Language Models with Reinforcement Learning](https://mlsanity.com/paper/2504.08837/) (2025)
- [Spatial-MLLM: Boosting MLLM Capabilities in Visual-based Spatial Intelligence](https://mlsanity.com/paper/2505.23747/) (2025)
- [MMSI-Bench: A Benchmark for Multi-Image Spatial Intelligence](https://mlsanity.com/paper/2505.23764/) (2025)
- [VLM-3R: Vision-Language Models Augmented with Instruction-Aligned 3D Reconstruction](https://mlsanity.com/paper/2505.20279/) (2025)
- [SpaceR: Reinforcing MLLMs in Video Spatial Reasoning](https://mlsanity.com/paper/2504.01805/) (2025)
- [SpatialBot: Precise Spatial Understanding with Vision Language Models](https://mlsanity.com/paper/2406.13642/) (2024)
- [3DSRBench: A Comprehensive 3D Spatial Reasoning Benchmark](https://mlsanity.com/paper/2412.07825/) (2024)
- [Reinforcing Spatial Reasoning in Vision-Language Models with Interwoven Thinking and Visual Drawing](https://mlsanity.com/paper/2506.09965/) (2025)
- [Cambrian-S: Towards Spatial Supersensing in Video](https://mlsanity.com/paper/2511.04670/) (2025)
- [Perception-R1: Pioneering Perception Policy with Reinforcement Learning](https://mlsanity.com/paper/2504.07954/) (2025)
- [MiMo-VL Technical Report](https://mlsanity.com/paper/2506.03569/) (2025)
- [MindCube: Spatial Mental Modeling from Limited Views](https://mlsanity.com/paper/2506.21458/) (2025)
- [SpatialLadder: Progressive Training for Spatial Reasoning in Vision-Language Models](https://mlsanity.com/paper/2510.08531/) (2025)
- [Visual Spatial Tuning](https://mlsanity.com/paper/2511.05491/) (2025)
- [Visionary-R1: Mitigating Shortcuts in Visual Reasoning with Reinforcement Learning](https://mlsanity.com/paper/2505.14677/) (2025)

## Abstract (arXiv)

Despite the remarkable prowess of Vision-Language Models (VLMs) in general multimodal tasks, they remain fundamentally ``flat'' when reasoning about the physical world. We argue that this spatial bottleneck stems from a profound dimensional mismatch: while VLMs are trained to interpret 2D projections, true spatial reasoning demands the recovery of latent 3D geometry and temporal continuity. To conquer this high-dimensional complexity, we advocate a shift from monolithic learning to a ``divide and conquer'' paradigm. We present FactoSR, a factorized reinforcement learning framework that explicitly interpret the dimensions collapsed by visual projection. At its core, FactoSR decomposes the monolithic problem of world-consistent reasoning into three orthogonal, geometric sub-objectives: planar correspondence (XY), depth consistency (Z), and temporal reversibility (T). By optimizing these verifiable constraints within a unified policy learning mechanism, we effectively transform an ill-posed projection recovery problem into a series of tangible reasoning steps. Extensive evaluations on multi-view and video benchmarks demonstrate that this elegant decomposition yields substantial gains in 3D and 4D reasoning, achieving a 5.9% boost on VSI-Bench and 4.5% on All-Angles-Bench. Our findings suggest that reinforcing explicit, factorized 4D consistency is a critical step toward evolving VLMs into robust, world-aware reasoners.