# Space Is Intelligence: Neural Semigroup Superposition for Riemannian Metric Generation

arXiv: https://arxiv.org/abs/2606.18828 · HTML: https://mlsanity.com/paper/2606.18828/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 24 references that are in the map.
- First posted on arXiv 17 Jun 2026.

## Builds on (most cited first, 11 of 11)

- [RT-2: Vision-Language-Action Models Transfer Web Knowledge to Robotic Control](https://mlsanity.com/paper/2307.15818/) (2023)
- [Diffusion Policy: Visuomotor Policy Learning via Action Diffusion](https://mlsanity.com/paper/2303.04137/) (2023)
- [π0: A Vision-Language-Action Flow Model for General Robot Control](https://mlsanity.com/paper/2410.24164/) (2024)
- [Octo: An Open-Source Generalist Robot Policy](https://mlsanity.com/paper/2405.12213/) (2024)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Genie: Generative Interactive Environments](https://mlsanity.com/paper/2402.15391/) (2024)
- [Learning Interactive Real-World Simulators](https://mlsanity.com/paper/2310.06114/) (2023)
- [Generalizing Convolutional Neural Networks for Equivariance to Lie Groups on Arbitrary Continuous Data](https://mlsanity.com/paper/2002.12880/) (2020)
- [Motion Planning Networks](https://mlsanity.com/paper/1806.05767/) (2018)
- [Riemannian Motion Policies](https://mlsanity.com/paper/1801.02854/) (2018)
- [NTFields: Neural Time Fields for Physics-Informed Robot Motion Planning](https://mlsanity.com/paper/2210.00120/) (2022)

## Built on by (most cited first, 1 of 1)

- [The Field Knows: Cross-Dimensional Geometry from Navigation to Black Holes](https://mlsanity.com/paper/2608.07566/) (2026)

## Abstract (arXiv)

Traditional approaches place intelligence in the agent, whether as a learned policy or a search procedure. We instead place intelligence in the space itself: a scene induces a Riemannian metric on the configuration manifold, and action reduces to following the geodesics of that metric rather than invoking a separate planner or collision checker. A single Encoder-Router network realizes this idea through three complementary parameter groups -- frame parameters that orient the generators, modulation parameters that govern their spatial propagation, and basic coefficients that determine their strength. These groups combine through a shared semigroup-superposition mechanism to produce a single Riemannian metric field, yielding a compact architecture whose geometry scales naturally with scene complexity. Trained on a single two-obstacle scene, the model demonstrates robust zero-shot generalization across unseen obstacle configurations, with orders-of-magnitude separation between collision-free and obstacle-penetrating path costs.