Cubical Agda Formalizes Topos Causal Models for Machine-Checked Inference

This paper introduces a groundbreaking machine-checked formalization of topos causal models, leveraging Cubical Agda. The work builds upon existing verified frameworks for probability and do-calculus, providing a robust foundation for causal inference. Key contributions include the formalization of interventions as characteristic maps, a proof of sheaf gluing for independent mechanisms, and a detailed machine-check of the internal language's Kripke-Joyal forcing clauses. A significant finding involves identifying and rectifying a gap in the standard Lawvere-Tierney axioms, which previously failed to enforce a closure operator. With this correction, the framework demonstrates that interventions and Pearl's causal rules remain stable across various topologies. This stability is crucial for understanding the transportability of counterfactuals across different regimes. Furthermore, the research extends the program by incorporating a machine-checked contextuality obstruction, revealing scenarios where locally consistent data cannot form a coherent global model. The entire development is axiom-free and type-checks safely, using rational numbers for concrete discharge, focusing on the presheaf fragment of the topos.