System-Level Invariants and Geometric Balance Relations in Dissipative Dynamics of Merging Domains: A Phenomenological Framework for Systems

Dissipative systems that evolve on time-dependent domains occur across systems science whenever redistribution, loss, and global restructuring are coupled with geometric change. This work develops a phenomenological, system-level framework for analyzing such processes and focuses on invariant organizational constraints rather than on microscopic mechanisms or specific physical realizations. Redistribution on an evolving domain is modeled through a diffusion–dissipation equation with curvature- and volume-dependent dissipative loss terms, interpreted as effective drivers of irreversible reorganization. Lie symmetry analysis reveals a non-semisimple structure whose generators act as invariants of admissible system-level reorganizations rather than as sources of conservation laws. By selecting a symmetry-compatible subalgebra, an emergent geometric representation is constructed that compactly encodes global balance constraints without invoking a physical spacetime interpretation. The framework yields time-independent geometric invariants and a system-level balance relation that stabilizes global organization despite ongoing local dissipation. A dimensionless geometric indicator is introduced to quantify intrinsic anisotropy of reorganization and to classify dissipative regimes. Owing to its invariant and phenomenological character, the approach is applicable to a broad class of complex systems with evolving domains and irreversible dynamics, consistent with the scope of systems research.