Ordinal Well-Foundedness as a Structural Termination Criterion for Recursive Systems

This preprint develops ordinal well-foundedness as a structural termination criterion for recursive systems. The central claim is that recursive computational processes, validation chains, proof dependencies, and layered oversight architectures can be evaluated by embedding their state evolutions into a countable well-founded ordinal hierarchy. By assigning each admissible state a canonical ordinal rank and requiring genuine reduction steps to be rank-decreasing, the framework excludes infinite descending dependency chains, strict-priority circularity, and procedural regress. The manuscript bridges foundational set theory and applied computational logic. It translates the set-theoretic Axiom of Foundation, ordinal rank, and cumulative hierarchy into an operational discipline for recursive digital processes. It defines an admissibility hierarchy, a canonical rank map, and transition classes including reduction, preservation, escalation, and discharge. The core termination results establish absence of infinite descending chains, universal termination for genuine reduction branches, and exclusion of strict-priority circular dependencies. The paper is positioned within the historical lineage of Zermelo’s axiomatic set theory, von Neumann’s ordinal-rank and cumulative-hierarchy perspective, well-founded orderings in termination analysis, and Mathias-adjacent foundational work on weak set theories, admissibility, recursive construction, and the critique of ungrounded formalism. In the contemporary application layer, the manuscript is associated with the Digital Fabrica / GILC research programme as a systems-level translation of ordinal well-foundedness into recursive-systems verification, validator discipline, and anti-regress architecture. This record is released as a preprint for scholarly documentation, public citation, and further review. It is not a journal-certified publication.