Inter‑Regime Law of Closures in Coherent Systems

This work introduces a universal structural law — the inter‑regime law of closures — showing that stable configurations emerging in disparate fields share a common mathematical origin. Magic numbers in nuclear physics, saturated combinatorial structures, attractors in coherent networks, and the privileged regimes of the active vacuum (RSU/CSU) all arise from the same mechanism: the interaction between an internal dynamics and an invariant defined on a coherent state space. The paper formalizes this mechanism using a minimal framework based on three elements: a state space 𝑿, an internal dynamics Φₜ, and an invariant 𝑰. We prove that any coherent system endowed with such an invariant naturally selects stable stationary states, called closures, characterized by stationarity, dynamic stability, and invariant saturation. Regimes — whether physical, combinatorial, cosmological, or dynamical — are shown to be classes of closures, not primitive assumptions. This provides a unified structural explanation for the emergence of stable patterns across domains traditionally considered unrelated. The law has direct implications for cosmology: in the CSU/RSU framework, the active vacuum behaves as a coherent system whose internal invariant selects stable cosmological closures. Cosmogony becomes a transition between closures rather than an absolute beginning, and multiple cosmological regimes arise as structural possibilities of the invariant. This work proposes a general theory of regimes, offering a common foundation for stability phenomena across physics, mathematics, and complex systems.