Topological Encoding of Verified Causal Structures into Continuous Epistemic Fields: A Sheaf-Theoretic Construction

Verified Causal Structures (VCS) — directed acyclic graphs with cryptographically attested edges representing independently verified causal dependencies — are inherently discrete objects. Continuous field representations of knowledge offer computational advantages including O(1) queries, natural multi-resolution analysis, and compatibility with geometric machine learning pipelines. We propose a sheaf-theoretic construction that achieves a principled hybrid: causal structure (edges, direction, compound verification depth) is encoded as topological invariants of a continuous epistemic field via DAG-indexed persistent homology and persistent path homology; cryptographic attestations (signatures, hashes) are carried by discrete sheaf stalks whose consistency is enforced by sheaf cohomology. We prove that this construction preserves all causal invariants of the original VCS, show that sheaf cohomological obstructions detect inconsistencies in the verification record, and argue that the resulting topological epistemic field is a more natural representation of verified knowledge than either the discrete DAG or a naive continuous embedding. Keywords: topological data analysis, persistent homology, cellular sheaves, verified causal structures, epistemic fields, knowledge representation, sheaf cohomology, causal inference, geometric AI