Quality-Operator Non-Collapse (QONC) introduces a certificate-first safety framework for recursive systems that repeatedly generate, evaluate, and reintegrate updates. The method is provenance-agnostic and relies only on observable, auditable quantities (confidence allocations, risk caps, controller certificates, and tamper-evident ledger debits), making it suitable for settings where source labels are unavailable or unreliable. The framework unifies: adaptive anytime candidate validity, nonstationary risk control with delayed-label correction and restart-safe stream inversion, viability-safe robust MPC, operadic and sheaf-cohomological composition for federated modules, and implementable topos/homotopy semantic monitoring. To preserve runtime liveness, semantic checks are compute-aware and queue/staleness-constrained, while hard confidence budgets are epochized and operational credits enable soft landing and recovery without weakening hard probabilistic guarantees. All claims are grounded in per-round observable failure tests and ledger-accounted debits, enabling deterministic replay, cryptographically verifiable lineage, and audit-ready deployment. Under stated assumptions, QONC provides anytime and compositional guarantees against quality collapse and liveness collapse in high-complexity recursive systems, including AI pipelines, autonomous scientific workflows, federated governance processes, and self-modifying software ecosystems.