General Cárdenas Theorem: Global Conservation, Local Openness, and Non-Identifiability of Effective Dynamics

The General Cárdenas Theorem establishes a formal framework for analyzing the relationship between global conservation laws and the effective dynamics of locally observed subsystems. Its central result is that global conservation does not necessarily imply local conservation, nor does it guarantee that the dynamics of an open subsystem can be uniquely reconstructed from its observable variables alone. Energy and other conserved quantities may be exchanged between interacting domains, while unobserved degrees of freedom and external couplings can influence the evolution of a locally accessible region. Consequently, a subsystem may exhibit an apparently non-conservative behavior even when the complete system obeys an exact global conservation law. The theorem is formulated through a general balance equation for interacting domains and is extended to continuous systems and relativistic energy-momentum conservation. A mechanical example involving two coupled degrees of freedom demonstrates that the total energy remains conserved while the energy assigned to an individual subsystem varies through interaction-mediated exchange. The analysis further establishes a condition of non-identifiability: distinct global states may produce identical local observations while generating different effective dynamical evolutions. Under such conditions, no unique autonomous law based exclusively on the available local observables can fully determine the subsystem's dynamics. The framework distinguishes three logically separate properties: global conservation, local openness, and local dynamical identifiability. It also discusses the implications of this distinction for physical modeling and cosmological descriptions, particularly when observations are restricted to a finite causal domain. The theorem does not prescribe a particular cosmological model; rather, it establishes a general structural limitation that any model based on incomplete local information must address.