The ψ₀-OCM Emergence Ladder: A Cross-Domain Redistribution Stability Derivation of Closure-Regulated Persistence

Persistent structures arise across nature at many scales, from cosmological structure formation to chemical self-organization and biological regulatory systems. Although these systems belong to different scientific disciplines, they frequently exhibit similar threshold behavior governing transitions between unstable and persistent regimes. Within the ψ₀-OCM (Osborne Cosmological Model), such transitions are interpreted as consequences of stabilization of redistribution flux originating in the primordial field ψ₀. In this work a microscopic stabilization law is introduced, a persistence functional is derived from redistribution dynamics, and a critical stabilization threshold is obtained from first principles. These elements lead to a closure–persistence theorem within the ψ₀-OCM framework that establishes a persistence inequality governing when structures remain stable against dissipation and boundary leakage. The resulting formulation naturally produces an emergence ladder linking cosmological, chemical, and biological systems through a common redistribution stability principle. Variable mappings are constructed across domains, connections are established between threshold operators used in astrophysics, chemical network theory, and biological regulation, and representative examples illustrate the operational application of the persistence condition. These results establish that many threshold mechanisms observed across scientific disciplines admit a common ψ₀-OCM persistence representation, appearing as domain-specific realizations of a shared redistribution stability condition.