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

2026-03-18

AI Summary

This work addresses the failure of local consistency assumptions in existing continuous generative causal models when confronted with nontrivial homological structures—such as structural conflicts or latent confounding. The authors formalize structural causal models as cellular sheaves over Wasserstein space and introduce a computable causal sheaf Laplacian by integrating entropy-regularized optimal transport with cohomological theory. They further propose an O(1) memory-efficient reverse-mode gradient algorithm. By unifying cellular sheaf theory, nonlinear Fokker–Planck dynamics, and Sinkhorn optimization, this study establishes the first sheaf-theoretic foundation for generative causal modeling. Applied to counterfactual inference in single-cell RNA sequencing, the proposed topological causal scoring significantly enhances sensitivity to complex causal structures, revealing that thermodynamic noise can traverse topological barriers via “entropic tunneling.”

Technology Category

• Machine Learning: Causal Learning
• Reasoning under Uncertainty: Causality
• Knowledge Representation and Reasoning: Action, Change, and Causality

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Social Networks and Social Media: Generative AI / large language models and their impact on social systems

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.

Problem

• counterfactuals
• causal models
• cohomological obstructions
• sheaf theory
• Wasserstein spaces

Innovation

• cohomological obstructions
• cellular sheaves
• entropic regularization
• Wasserstein causal Laplacian
• topological causal discovery