Open-System Geometric Flows in Dimension Four: Dissipation, Admissibility, and Analytic Control

This paper develops Implicate Fluid Geometrodynamics (IFG) as an open-system framework for four-dimensional geometric evolution. We formulate the IFG-DeTurck system as a strictly parabolic modification of Ricci–DeTurck flow coupled to an auxiliary dissipative sector. The foundational analytic structure is developed, including well-posedness, gauge fixing, Lyapunov dissipation identities, and compatibility with parabolic regularity theory. The paper establishes IFG as a flexible analytic platform for studying singularity formation, dissipation, and stability in critical dimension four. Future works will refer to this as Paper I.