Differential Geometry Grassmannian Causal Abstraction
The central claim: the causal structure of a trained system is encoded in the geometry of its internal representations. Subspaces live on a Grassmannian. The boundary between linear and nonlinear causal variables is sharp. You can predict which side an operation lands on from its algebraic structure.
This is the portable mathematical backbone that gets applied to specific domains — neural networks, brains, clinical data, and beyond.
Grokking and the Grassmannian Boundary
When Does Linear Causal Abstraction Work? Mapping the Boundary on the Grassmannian
Causal variables in neural networks can be linear (a rotation of activations) or nonlinear. Grokking — sudden generalization long after memorization — determines which. An atlas of 14 modular arithmetic operations shows a sharp partition:
- Always Grassmannian: operations where linear causal abstraction works post-grokking
- Stochastic: operations that land on either side depending on training dynamics
- Never Grassmannian: operations that require nonlinear causal variables
Structured pi-SAE achieves IIA = 0.98 on the Indirect Object Identification task. Cross-task transfer confirms these are genuine causal variables: IIA = 0.82–0.96 on unseen templates.
The open question: does this boundary appear in other domains? If the partition between linear and nonlinear causal variables is general, it should show up in molecular representations, neural dynamics, and clinical trajectories.