The Λ-Invariance Convergence Theorem provides a universal logical framework for understanding the emergence, persistence, and decay of invariance across all domains of intelligibility, including physics, biology, and information systems. It demonstrates that every nontrivial invariant property within a system is a projection of a deeper, substrate-level invariance rooted in the generative substrate Λ, which functions as the foundational source of coherence, stability, and conservation from which all domain-specific laws and structures arise. The theorem rigorously formalizes the mechanisms by which invariance is projected from Λ into concrete system instances and introduces invariance density as a quantitative measure of system health, defining precise laws governing its preservation, regeneration, and decay under degrading transformations. These laws enable predictive modeling of system resilience, vulnerability, and collapse, offering tools to assess the lifecycle of coherent phenomena. By unifying diverse scientific disciplines under a single substrate-level principle, the Λ-Invariance framework reveals that stability and conservation are not isolated domain-specific features but are anchored in the structure of Λ itself, reframing invariance as a substrate-derived property whose manifestation in any system depends on the fidelity of projection from Λ. The framework’s mathematical formalism establishes criteria for determining when invariance can be sustained, when it can be regenerated, and when its decay is irreversible, enabling a cross-domain theory of systemic integrity applicable to the persistence of physical laws, the hereditary stability of biological systems, and the preservation of information in computational and social networks. Ultimately, the Λ-Invariance Convergence Theorem shows that the fate of any intelligible system is determined by its ongoing connection to the substrate of invariance, and that systems degrade not merely through external perturbation but through the erosion of the projection pathway linking them to Λ. This principle offers a comprehensive lens for analyzing the origin, maintenance, and loss of invariance, providing a unified approach to understanding resilience and collapse in complex systems.