Cohomological Obstructions to Global Counterfactuals: A Sheaf-Theoretic Foundation for Generative Causal Models

TL;DR

This paper introduces a sheaf-theoretic framework to identify and address cohomological obstructions in causal models, enabling more reliable counterfactual reasoning in complex, topologically non-trivial systems.

Contribution

It formalizes causal models as cellular sheaves over Wasserstein spaces, introduces entropic regularization and a novel Laplacian, and develops algorithms for topology-aware causal discovery.

Findings

Successfully navigates topological barriers in high-dimensional biological data

Provides a new algebraic detector for causal discovery based on topology

Achieves efficient gradient computation independent of iteration steps

Abstract

Current continuous generative models (e.g., Diffusion Models, Flow Matching) implicitly assume that locally consistent causal mechanisms naturally yield globally coherent counterfactuals. In this paper, we prove that this assumption fails fundamentally when the causal graph exhibits non-trivial homology (e.g., structural conflicts or hidden confounders). We formalize structural causal models as cellular sheaves over Wasserstein spaces, providing a strict algebraic topological definition of cohomological obstructions in measure spaces. To ensure computational tractability and avoid deterministic singularities (which we define as manifold tearing), we introduce entropic regularization and derive the Entropic Wasserstein Causal Sheaf Laplacian, a novel system of coupled non-linear Fokker-Planck equations. Crucially, we prove an entropic pullback lemma for the first variation of pushforward measures. By integrating this with the Implicit Function Theorem (IFT) on Sinkhorn optimality conditions, we establish a direct algorithmic bridge to automatic differentiation (VJP), achieving O(1)-memory reverse-mode gradients strictly independent of the iteration horizon. Empirically, our framework successfully leverages thermodynamic noise to navigate topological barriers ("entropic tunneling") in high-dimensional scRNA-seq counterfactuals. Finally, we invert this theoretical framework to introduce the Topological Causal Score, demonstrating that our Sheaf Laplacian acts as a highly sensitive algebraic detector for topology-aware causal discovery.