A Formally Verified Acyclic Derivation Order for a 10,271-Node Unification Registry: Dependency Topology, Depth Metrics, and an Isabelle/HOL Certification Pathway

Description This paper (Paper X in the TZPID Proof Pipeline Series) documents a critical registry-level dependency breakthrough in the TZPID proof pipeline. The canonical unification registry contains 10,271 entries, which have been systematically reorganized from a flat list into a strict, dependency-respecting proof ladder to establish a concrete derivation order. By checking the materialized rung-to-rung dependencies using an Isabelle/HOL Sledgehammer harness, the package proves graph non-circularity: every prerequisite entry strictly precedes its dependent entry ($d < r$). This structural verification ensures that downstream computational certificates, proof files, and peer-review artifacts remain completely acyclic and mathematically self-consistent. Key Metrics & Topological Findings Total Registry Nodes: 10,271 entries. Materialized Dependency Edges: 753 concrete edge cases formally certified in Isabelle/HOL. Parallelization Bottleneck (Occam-depth): Maximum sequential dependency depth of 21 edges, enabling parallel processing across 22 sequential layers rather than 10,271 serial steps. Isolated vs. Non-Isolated Nodes: 9,460 isolated entries (0 incoming/outgoing edges) and 811 non-isolated nodes. Risk Zones: 138 nontrivial connected clusters, with the largest component spanning 330 nodes and 385 edges. Linearization Handling: 763 prose-level dependency edges identified, with 9 cycle-partner edges strategically cut for linearization and retained for future refinement. Included Artifact Replicas & References The audit spans several interconnected files and verification frameworks preserved within the publication package, including: TZPID_ALL_IDS_REASSEMBLED_PROOF_LADDER.md (The derivation-order authority). graph_topology/materialized_edges.csv (The 753 concrete edge cases). sledgehammer/TZPID_Proof_Ladder_Sledgehammer.thy (The Isabelle/HOL order-checking harness). graph_topology/GRAPH_TOPOLOGY_REPORT.md (Connected components and super-node metrics).