用单一损失训练的连续度量场，跨维度发现从导航到黑洞的几何结构。
The Field Knows: Cross-Dimensional Geometry from Navigation to Black Holes
- 基于因果对比损失，将场景编码为对称矩阵系数并生成黎曼或洛伦兹度量。
- 零样本泛化验证了其捕捉可迁移几何结构的能力，非记忆特定配置。
- 同一框架在不同维度下自动生成黑洞事件视界等物理级几何现象。
我们提出一种由单一因果对比损失训练的连续度量场框架。该框架将场景编码为固定对称矩阵基的系数，组合成李代数元素，并指数映射为黎曼或洛伦兹度量。在不同维度中，该场发现了完整的几何结构谱：从平面和机械臂配置空间中的避障测地线，到洛伦兹时空中的黑洞事件视界。广泛的零样本泛化研究证明，该场捕捉的是可迁移的几何结构而非记忆特定配置。在黑洞设定中，因果损失自发演化出具有正确洛伦兹符号的类黑洞结构。相同的损失、架构与训练协议，在跨维度下生成了全部几何现象。场知几何，几何亦知物理。
原文摘要 · Abstract (English)
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.
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