Causal Schrödinger Bridges: Constrained Optimal Transport on Structural Manifolds

Generative modeling typically seeks the path of least action via deterministic flows (ODE). While effective for in-distribution tasks, we argue that these deterministic paths become brittle under causal interventions, which often require transporting probability mass across low-density regions ("off-manifold") where the vector field is ill-defined. This leads to numerical instability and spurious correlations. In this work, we introduce the Causal Schrödinger Bridge (CSB), a framework that reformulates counterfactual inference as Entropic Optimal Transport. Unlike deterministic approaches that require strict invertibility, CSB leverages diffusion processes (SDEs) to robustly "tunnel" through support mismatches while strictly enforcing structural admissibility constraints. We prove the Structural Decomposition Theorem, showing that the global high-dimensional bridge factorizes into local, robust transitions. Empirical validation on high-dimensional interventions (Morpho-MNIST) demonstrates that CSB significantly outperforms deterministic baselines in structural consistency, particularly in regimes of strong, out-of-distribution treatments.