Viable Continuation Under Constraint A General Theory of Stability, Pathology, and Collapse Across Reflexive Systems

We introduce a general theorem framework for systems whose persistence depends on reconciling local transitions with whole-system viability. The framework is intentionally abstract: it applies to record-bearing, recursively constrained systems equipped with transitions, viability conditions, traces, channels, anchors or proxies, and a local-to-global compatibility relation. We then develop a theorem spine in three levels. First, a family of foothill theorems shows that local structural defects induce failure of corresponding system-level robustness properties. Second, a family of ridge theorems composes these failures into broader structural impossibility results concerning proxy guidance, local admissibility, correction through multiplicity, and viability support through sufficient capacity. Third, a summit abstraction, BoundaryDefect, unifies the main defect families and yields a general viability-boundary theorem: whenever a system exhibits any one of the identified boundary defects, it fails a corresponding schema of robust viable continuation. The abstract theorem family is fully proved in Lean; the domain sections that follow are interpretation schemas and corollary schemas, not full domain formalizations. The framework is machine-checked and is designed to support disciplined instantiation across diverse domains, including self-modifying AI systems, legal and institutional orders, organismal and ecological systems, markets, civilization-scale coordination systems, and other recursively constrained structures. Trust boundary. Machine-checked claims are those in -continuation-lean matching. Domain vignettes and bridge narratives are schemas, not additional closed theorems unless a bridge module proves a toy witness. Summit anchors: conditionsforᵣobustᵥiablecontinuation; ᵥiabilityboundary.