Topological Encoding of Verified Causal Structures into Continuous Epistemic Fields: A Sheaf-Theoretic Construction
↓ Download PDFAbstract. We construct a sheaf-theoretic encoding of a Verified Causal Structure as a continuous epistemic field, with cryptographic attestations carried in discrete sheaf stalks. The construction allows the topology of verified knowledge — its connected components, holes, voids, and persistent features — to be analyzed using the full machinery of topological data analysis while preserving the discrete verification semantics of the underlying VCS.
1. The Sheaf
We define a sheaf F over the VCS where each open set carries the joint attestation context, and stalks carry the cryptographic certificates themselves.
2. Continuous Epistemic Field
Sheaf cohomology induces a continuous field of epistemic confidence over claim-space.
3. Persistent Homology
Persistent features of the field correspond to robust knowledge clusters; transient features correspond to under-attested or contested claims.
4. Applications
Topological invariants enable detection of knowledge gaps, redundant verification, and structural fragility in the verified knowledge graph.
5. Conclusion
The sheaf encoding bridges discrete cryptographic verification with continuous geometric analysis of knowledge.
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