The Primeval Atom: Covariant Quantum Hydrodynamic Regularization and Ultraviolet Bounding in \(\mathbb{R}^{3}\).

This technical memorandum presents a unified ultraviolet regularization model for energy density in high-energy regimes using the covariant Madelung-Klein-Gordon hydrodynamic formalism. Through functional analysis, we rigorously demonstrate that introducing a critical geometric radius R = sqrt(pi) in phase space operates as an orthogonal projector in Fourier space. This mechanism effectively interrupts the ultraviolet divergence of enstrophy within Sobolev spaces H^s(R^3). Furthermore, we analytically derive a stationary libration point characterized by the total collapse of the material derivative of the local velocity field to zero, thereby immunizing the system against global entropic degradation. The internal consistency of the model is validated via computational numerical simulation.