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

Rui Wu, Hong Xie, Yongjun Li

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.

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

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.
Paper Structure (44 sections, 11 theorems, 61 equations, 5 figures, 2 tables, 1 algorithm)

This paper contains 44 sections, 11 theorems, 61 equations, 5 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction and Motivating Example
  2. Related Work
  3. Mathematical Preliminaries
  4. Wasserstein Geometry and Otto Calculus
  5. Regularity Assumptions
  6. Cellular Sheaves and Cohomology
  7. The Entropic Causal Sheaf and Topological Linearization
  8. Stalks, Restriction Maps, and Metric Sheaf Discrepancy
  9. Linearization: The Tangent Causal Sheaf and Rigorous H1
  10. The Entropic Pullback Lemma and Sheaf Laplacian
  11. Topological Frustration and Global Equilibrium
  12. The Topological Frustration Inequality
  13. Linearization on the Tangent Sheaf: The Wasserstein Hodge Decomposition
  14. The Cohomological Necessity of Causal Equilibrium
  15. Strict Energy Dissipation via Metric Gradient Flows
  16. ...and 29 more sections

Key Result

Lemma 4

Let $T: X \to Y$ be a smooth diffeomorphism, $\nu \in \mathcal{P}_2(Y)$ a fixed target measure, and $F_\varepsilon(\mu) = \frac{1}{2}\mathcal{W}_{2, \varepsilon}^2(\nu, T_{\#} \mu)$. The first variation (Wasserstein gradient) of $F_\varepsilon$ evaluated at the source measure $\mu$ is strictly given where $g^{(\varepsilon)}: Y \to \mathbb{R}$ is the unique Sinkhorn dual potential mapping $T_{\#}\m

Figures (5)

  • Figure 1: The impact of topological obstructions on causal generative models. (a) Unregularized, deterministic models attempt to resolve the contradiction by pushing probability measures to infinity, resulting in catastrophic manifold tearing. (b) Our proposed Entropic Sheaf Laplacian leverages thermodynamic diffusion to gracefully resolve the conflict, guiding the system to a globally coherent stationary state.
  • Figure 2: High-resolution vector field of the 2D Wasserstein Sheaf Flow over 500 steps, overcoming orthogonal topological conflicts.
  • Figure 3: Algorithmic Benchmarks (IFT vs. Naive Unrolling).(Left): Gradient computation time per backward pass. The IFT explicitly avoids traversing the $L$-step computational graph, achieving significant acceleration. (Right): Reverse-mode memory footprint. Naive unrolling suffers from catastrophic $\mathcal{O}(L \cdot N^2)$ linear explosion, effectively prohibiting high-dimensional deep learning. In stark contrast, our IFT-VJP formulation dynamically strictly bounds the memory strictly to $\mathcal{O}(N^2)$, rendering the algorithmic footprint completely invariant to the Sinkhorn horizon.
  • Figure 4: Counterfactual Intervention on PBMC 3k scRNA-seq. The Naive ODE (dashed gray) fails by entering the zero-density void (Biological Chimera). The GACF Path (solid red) autonomously navigates the manifold, utilizing entropic tunneling to overcome the topological frustration and reach the target Monocyte cluster.
  • Figure 5: Evolution of the Topological Causal Score $\mathcal{S}(\mathcal{G})$ during the Entropic Sheaf Flow. The Dirichlet energy of the true graph converges to the thermodynamic floor, whereas the spurious graph is permanently trapped by the $H^1$ topological barrier, perfectly verifying Theorem \ref{['thm:spurious_edges']}.

Theorems & Definitions (15)

  • Remark 1: Relaxation of Bi-Lipschitz Diffeomorphisms in Deep Learning
  • Definition 2: Variational Metric Obstruction
  • Definition 3: Strict Cohomological Obstruction
  • Lemma 4: First Variation via Entropic Pullback
  • Theorem 5: The Entropic Wasserstein Sheaf Laplacian
  • Lemma 6: Equivalence of Variational and Cohomological Obstructions
  • Theorem 7: Topological Frustration Inequality
  • Theorem 8: Wasserstein Causal Hodge Decomposition
  • Theorem 9: Cohomological Necessity of Causal Equilibrium
  • Theorem 10: Energy Dissipation Identity
  • ...and 5 more