Formal Pressure Testing a Relation-First Ontology: Representability, Non-Canonicity, and Structural Limits - The Universal Recursive Dynamics Case

This release presents an Open Specialist Assessment Version of a formal pressure test concerning selected commitments of Universal Recursive Dynamics (URD) and categorical, dependent, history-sensitive, and process-theoretic resources used to represent them without collapsing structural conditioning into temporal succession, constitution into restriction, or participant continuation into current-state identity. The investigation originated when the author recognized a possible structural correspondence between unresolved URD questions and split-idempotent/Karoubi resources encountered in Hefford and Wilson's work on hyper-decoherence from a causally indefinite process theory to quantum theory. The author is not a specialist in the mathematical fields employed and could not independently determine whether that apparent correspondence was mathematically substantive. Sustained human–AI inquiry was therefore used to translate the initial question into explicit definitions, candidate constructions, propositions, proofs, counterexamples, negative results, and unresolved formal obligations. The manuscript reports both positive and negative results. Split-idempotent/Karoubi machinery is retained as a resource for restriction and invariance but does not supply relation-priority or constitution; compatible-idempotent order does not by itself encode Paper-A structural conditioning. Dependent/displayed structures provide a bounded representation of specifiability dependence and display-relative polarity; an inductive-inductive witness supplies a limited joint-generation capability and a paper-local specification non-separability result. History-sensitive individuation, register, and continuation resist reduction to endpoint or current state, while the proposed Part III interface remains representational rather than derivational and retains explicit open obligations. The surviving formal material is released for independent specialist scrutiny. AI-assisted convergence is not treated as a substitute for disciplinary verification, and the author does not claim specialist mathematical competence sufficient to certify the formal results independently. In this assessment version, PROVED means that a result is claimed to follow from the stated assumptions by the proof or argument supplied, or is a directly used standard result; it does not imply prior independent specialist verification. Corrections, counterexamples, typing objections, provenance challenges, scope objections, and identification of prior mathematical formulations are specifically invited. A separate Technical Assessment Guide accompanies the manuscript so that readers may evaluate only the portions within their expertise.