Mathematical models used in hydrology, physics, chemistry, and many engineering systems are largely based on empirical or semi-empirical relationships. One fundamental reason for this is that classical coordinate systems—whether Cartesian or relativistic spacetime—are inherently incapable of representing the laws of conservation in open systems. These frameworks describe the instantaneous state of quantities, but do not inherently incorporate the flow, input–output balance, and irreversible nature of processes into their geometric structure. In this paper, a new geometric–physical framework for modeling open systems is introduced, based on a four-dimensional manifold with a structural axis of equilibrium L and the phenomenon–time law. In this framework, the conserved quantities are not simply described by spatiotemporal coordinates, but their evolution path, input flow, and equilibrium state are represented explicitly along the L-axis. Thus, the conservation law is applied directly to the geometry of the system state space, not as an external constraint. The proposed framework separates the dynamics of open systems into three general components: initial irreversible acceptance, two-way equilibrium acceptance, and residual phenomena. The geometric-physical coordinate framework introduces an analytical tool for the geometric representation and structural comparison of the behavior of open systems under known non-equilibrium thermodynamics. Unlike conventional empirical approaches, the present framework allows for the analytical determination of capacities, equilibrium thresholds, and limit behaviors of the system without heavy reliance on experimental calibration. The results show that the phenomenon-time law can be used as a unified geometric foundation for modeling open systems under survival laws and provides a coherent platform for theoretical analysis, dynamic simulation, and interdisciplinary generalization.
S. Shamohammadi (Wed,) studied this question.