UNIVERSAL LAW OF SYSTEMS Admissibility, Reachability, and the Geometry of Survival

AbstractWe introduce a universal structural law governing the viability of complex systemsacross mathematical, physical, computational, and civilizational domains. The centralclaim is that system survival is not determined by strength, performance, or localstability, but by the extent and accessibility of admissible states under boundedgenerative and search capacities. Let the total state-space of a system, andT admissible subset required for continued non-terminal operation. LetReach (S) denote the set of states reachable under the system’s generative structure and boundedsearch dynamics. We establish the fundamental condition of viability: T Reach (S) A system remains viable if and only if all admissible states remain reachable. Collapse isdefined as the loss of this condition, i. e. , the emergence of admissible but unreachablestates. We further show that viability depends on three irreducible quantities: (1) Admissibility — the survival-constrained domain of states, (2) Closure — the generative capacity to produce admissible configurations, (3) Search — the operational ability to identify and stabilize viable trajectories underbounded time and coordination. This yields the universal law: A system remains viable only while its reachable admissible state-space remainssufficiently complete, connected, and discoverable. The framework unifies problems traditionally treated as domain-specific—includingadditive closure in mathematics, controllability in dynamical systems, institutionalresilience, and civilizational continuity—under a single structural condition. This formulation transforms stability from a local dynamical property into a globalreachability constraint, providing a rigorous basis for analyzing fragility, collapse, andthe limits of complex order.