Exact Solution to the Galactic Halo Problem via Madelung Hydrodynamic Formalism.

This research presents the definitive analytical solution to the density divergence problem in galactic halos, widely known as the "cusp-core" problem. By transitioning from the Schrödinger-Poisson framework to the Euler-Madelung hydrodynamic description, this work demonstrates that the intrinsic quantum pressure of the dark matter fluid acts as a fundamental stabilization mechanism. The author, Armando Núñez Martínez, provides a rigorous step-by-step derivation showing that the Bohm quantum potential generates a localized repulsive barrier that grows faster than classical gravitational attraction as r \to 0. This physical constraint necessitates that the density gradient vanishes at the origin, effectively eliminating the singularity predicted by classical \LambdaCDM models. Key Results: Singularity Elimination: Mathematical proof that the Bohm potential prevents gravitational collapse into an infinite density point. Mathematical Necessity: The existence of a finite central density core (\rho_c) is established as a necessary consequence of the stabilization of self-gravitating quantum fluids. Field Unification: The study confirms that physical infinities are mathematically excluded from the architecture of the universe when wave nature and quantum effects are properly accounted for via the Euler-Madelung formalism. Keywords: Dark Matter, Galactic Halos, Quantum Hydrodynamics, Madelung Transformation, Cusp-Core Problem, Bohm Potential, Mathematical Physics.