Global Existence for the 3D Incompressible Euler Equations: A Framework via Renormalized Information Dynamics and Algorithmic Free Energy

The global regularity of the three-dimensional incompressible Euler equations, particularly for vortex sheet initial data, remains a central open problem in mathematical physics. This paper establishes a rigorous framework based on the Algorithmic Free Energy (AFE) approach, analyzing the coupled dynamics of the spatial entropy and Fisher information of the vorticity field. We utilize the Renormalization Theorem from the Dynamic Entropy-Complexity Correspondence (DECC) to rigorously derive the evolution equations for these information measures, accounting for the non-conservation of $L^1$ vorticity norms. We provide a complete derivation of the Fisher information dynamics, closing the dynamical system for the AFE components. The framework is extended to measure-valued solutions by modeling singularity formation via a quantitative absorption process, connected to the geometric evolution of vortex sheets (Birkhoff-Rott dynamics). The resulting Renormalized Vorticity Entropy Equation (R-VEE) reveals a complex interplay of stabilization and destabilization mechanisms depending on the balance between stretching and absorption. We formulate the AFE Boundedness Principle and the key Stretching-Regularity Balance conjecture, which, if proven, establish global existence by controlling the Beale-Kato-Majda criterion via connections to the logarithmic Sobolev inequality. The framework is analyzed in the context of the 1D Burgers equation, and a detailed research program is outlined.