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

2026-08-03

AI Summary

This work proposes a unified continuous metric field framework capable of automatically learning and generating appropriate geometric structures across diverse dimensional settings, ranging from robotic navigation to black hole spacetimes. The approach encodes scenes as coefficients over a symmetric matrix basis, combines them into Lie algebra elements, and constructs Riemannian or Lorentzian metrics via the exponential map, trained solely with a single causal contrastive loss. For the first time, it achieves zero-shot generalization across dimensional geometric tasks using a single architecture, loss function, and training protocol, spontaneously emerging black hole–like structures with correct Lorentzian signature without explicit supervision. Experiments demonstrate that the framework accurately recovers true geometries in both robotic obstacle avoidance and black hole event horizon modeling, validating geometric metrics as a unifying language for describing diverse physical phenomena.

Abstract

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.

Problem

• metric field
• cross-dimensional geometry
• geodesics
• black holes
• Lorentzian spacetime

Innovation

• metric field
• causal contrastive loss
• Lie algebra
• zero-shot generalization
• Lorentzian geometry