# The Field Knows: Cross-Dimensional Geometry from Navigation to Black Holes

arXiv: https://arxiv.org/abs/2608.07566 · HTML: https://mlsanity.com/paper/2608.07566/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 10 references that are in the map.
- First posted on arXiv 3 Aug 2026.

## Builds on (most cited first, 7 of 7)

- [Motion Planning Networks: Bridging the Gap Between Learning-based and Classical Motion Planners](https://mlsanity.com/paper/1907.06013/) (2019)
- [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)
- [RMPflow: A Geometric Framework for Generation of Multi-Task Motion Policies](https://mlsanity.com/paper/2007.14256/) (2020)
- [Reactive Motion Generation on Learned Riemannian Manifolds](https://mlsanity.com/paper/2203.07761/) (2022)
- [Dynamic Optimization Fabrics for Motion Generation](https://mlsanity.com/paper/2205.08454/) (2022)
- [Space Is Intelligence: Neural Semigroup Superposition for Riemannian Metric Generation](https://mlsanity.com/paper/2606.18828/) (2026)

## Abstract (arXiv)

We introduce a continuous metric field framework trained by a single causal contrastive loss. The framework encodes a scene into coefficients of a fixed symmetric matrix basis, assembles them into a Lie algebra element, and exponentiates the result to a Riemannian or Lorentzian metric. Across dimensions, this field discovers the full spectrum of geometric structures: from obstacle-avoiding geodesics in robot navigation across planar and manipulator configuration spaces, to event horizons of black holes in Lorentzian spacetime. Extensive zero-shot generalization studies demonstrate that the field captures transferable geometric structure rather than memorizing specific configurations. In the black hole setting, the causal loss spontaneously evolves genuine black-hole-like structures with the correct Lorentzian signature. The same loss, the same architecture, and the same training protocol produce the full range of geometric phenomena across dimensions. The field knows geometry, and geometry knows physics.