Space as an Interventional Invariant: Cross-Modal Predictive Geometry for Stratified Cities and Em-Spaced Intelligence

The paper proposes defining space as an interventional invariant—a minimal relational structure that maintains local compatibility and future observation laws under admissible actions. It introduces a cross‑modal predictive geometry that combines local state spaces, modality‑specific observation maps, an action groupoid, and a predictive‑state quotient, with causal conditions to distinguish interventional from observational structure. The authors prove that, given joint point separation, equivariance, and interventional faithfulness, the latent space is identifiable up to the centraliser of the intervention group, reducing ambiguity to residual coordinate freedom, and extend the framework to stratified urban systems using sheaf‑valued representations.

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• diffusion
• robotics
• multimodal