Observable Continuity - A Hierarchical Framework Based on Admissibility, Persistence, Transmission, and Adaptive Dynamics

Physical systems exhibit an extraordinary diversity of adaptive behaviors. From quantum control and information processing to biological evolution and artificial intelligence, adaptive dynamics repeatedly emerge despite the enormous differences among the underlying physical mechanisms. Current theories successfully describe the laws governing these systems, yet no general framework explicitly organizes how physically admissible trajectories become persistent, how persistent structures achieve long-term continuity through transmission, and how adaptive processes operate within this hierarchy. This work proposes a theoretical framework in which observable continuity is interpreted as the final consequence of a nested hierarchy of physical constraints. The framework is developed within the conceptual architecture of the CLEO–LOP–COUT research program, where admissibility determines which trajectories may exist, persistence determines which admissible trajectories remain operational, and transmission determines which persistent organizations survive transitions between successive physical carriers. Adaptive dynamics are introduced not as an additional physical law but as an operational mechanism exploring the admissible region defined by existing physical constraints. A mathematical hierarchy of trajectory spaces is proposed, together with admissibility, persistence, and transmission functionals. Observable continuity is represented by a generalized continuity functional whose simplest factorized approximation reproduces the product of admissibility, persistence, and transmission. The resulting formalism naturally leads to a continuity evolution equation, a hierarchy of admissible subsets, and a series of quantitative predictions that can be investigated through independent mathematical analysis, numerical simulations, and experimental tests. The framework does not modify quantum mechanics, thermodynamics, statistical mechanics, or information theory. Instead, it provides an organizational language intended to describe adaptive physical processes within the limits already imposed by those theories. Entropy-limited quantum reset is presented as one possible application of the formalism rather than as its foundation. The proposed theory should therefore be regarded as a mathematically structured hypothesis whose scientific value depends entirely upon future theoretical development and empirical validation.