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

arXiv:2609.11959v1 Announce Type: new Abstract: Space is a foundational concept across mathematics, physics, spatial cognition, urban science, and embodied intelligence, yet these fields often treat spatial structure either as a shared geometric container or as a collection of disconnected representations. Such approaches struggle to explain how heterogeneous sensory and urban processes can jointly reveal a common spatial structure, particularly when different modalities do not share the same metric or representation. This paper addresses this gap by defining space as an interventional invariant: the minimal relational structure that preserves local compatibility and the conditional laws of future observations under admissible actions. We develop a cross-modal predictive geometry that integrates local state spaces, modality-specific observation maps, an action groupoid, and a canonical predictive-state quotient, with explicit causal conditions for identifying interventional rather than merely observational structure. The key theoretical result shows that, under joint point separation, equivariance, and interventional faithfulness, the latent space is identifiable up to the centraliser of the intervention group, thereby reducing representational ambiguity to residual coordinate freedom. The framework is further extended to stratified urban systems using sheaf-valued representations, allowing geometric, physical, mobility, social, and economic layers to coexist without being reduced to a single metric. Synthetic experiments under noise evaluate equivariance, predictive suffic

No substantive claim about AI's effect on society.

The paper presents a theoretical and technical framework that could unify understanding of space across diverse fields, potentially improving AI systems and urban analysis. However, it is primarily conceptual with synthetic experiments but lacks direct evidence of broad societal impact. The implications are potentially positive for embodied AI and urban science, but practical benefits and risks remain uncertain.

This is abstract mathematical theory about spatial representation learning, tested only on synthetic experiments. It has no demonstrated societal application yet and no clear near-term implication for human flourishing either way.

This academic paper describes a theoretical framework for representing spatial structures in AI, particularly for urban systems. It focuses on mathematical and computational advancements without making claims about direct societal implications for human flourishing.

The paper presents a theoretical framework for integrating spatial structures across different modalities. It does not make substantive claims about AI's effect on society.

SOURCE     arXiv cs.LG (Machine Learning) (tier 1)
   ↓
DOCUMENT   59ea9481-d991-4e97-9204-a0485c3969e5
           https://arxiv.org/abs/2609.11959
   ↓
EVIDENCE   1 extracted excerpt
   ↓
MODEL RUN  5 runs, methodology 1.0.1
   ↓
SCORE      -29  (Adverse)
   ↓
CONFIDENCE 39%