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 the pathology of anticipatory control. In this work, we introduce the Causal Schrodinger Bridge (CSB), a framework that reformulates counterfactual inference as Entropic Optimal Transport. By leveraging diffusion processes (SDEs), CSB enables probability mass to robustly "tunnel" through support mismatches while strictly enforcing structural admissibility. We prove the Structural Decomposition Theorem, showing that the global high-dimensional bridge factorizes exactly into local, robust transitions. This theorem provides a principled resolution to the Information Bottleneck that plagues monolithic architectures in high dimensions. We empirically validate CSB on a full-rank causal system (d=10^5, intrinsic rank 10^5), where standard structure-blind MLPs fail to converge (MSE ~0.31). By physically implementing the structural decomposition, CSB achieves high-fidelity transport (MSE ~0.06) in just 73.73 seconds on a single GPU. This stands in stark contrast to structure-agnostic O(d^3) baselines, estimated to require over 6 years. Our results demonstrate that CSB breaks the Curse of Dimensionality through structural intelligence, offering a scalable foundation for high-stakes causal discovery in 10^5-node systems. Code is available at: https://github.com/cochran1/causal-schrodinger-bridge

翻译：生成建模通常通过确定性流（ODE）寻求最小作用量路径。尽管对于分布内任务有效，我们认为这些确定性路径在因果干预下会变得脆弱，因为因果干预通常需要将概率质量跨越向量场未定义的低密度区域（“离流形”）进行输运。这导致数值不稳定性和前瞻性控制的病态问题。在本工作中，我们引入了因果薛定谔桥（CSB），该框架将反事实推断重新表述为熵最优输运问题。通过利用扩散过程（SDE），CSB使概率质量能够鲁棒地“隧穿”支撑集不匹配区域，同时严格保证结构可容许性。我们证明了结构分解定理，表明全局高维桥可精确分解为局部、鲁棒的转移过程。该定理为困扰高维单体架构的信息瓶颈问题提供了原则性解决方案。我们在一个满秩因果系统（d=10^5，本征秩10^5）上实证验证了CSB，其中标准的结构无关MLP无法收敛（MSE ~0.31）。通过物理实现结构分解，CSB在单GPU上仅用73.73秒即实现了高保真度输运（MSE ~0.06）。这与结构无关的O(d^3)基线方法形成鲜明对比，后者估计需要超过6年时间。我们的结果表明，CSB通过结构智能打破了维度诅咒，为10^5节点系统中的高风险因果发现提供了可扩展的基础。代码发布于：https://github.com/cochran1/causal-schrodinger-bridge