Vortex Stability and Structure in the PEW Framework: Rigorous Foundations, Self-Adjoint Extensions, and the Hierarchical Dance of Cosmic Vortices (Article 4)

The Primordial Energy Wave (PEW) framework models all physical entities as topological vortex configurations of a single complex scalar field Ψ = ÷exp(iθ). Previous articles established the Lagrangian, derived the gravitational constant (κ²=8π), and computed the fine-structure constant (α_em = 1/137.018, within 0.013% of CODATA). The present article addresses two foundational questions left open: why do these vortex solutions persist, and what is the precise geometric structure of their cores?We first formalise the boundary conditions at the vortex core (ρ→0) using the theory of self-adjoint extensions of singular differential operators, drawing on the Infinitesimally Punctured Geometry (IPG) framework of Smarandache (2026). We show that the radial profile ODE of PEW near ρ=0 belongs exactly to the class of operators studied by IPG: the core is a measure-zero defect carrying distributional curvature, and the physical boundary condition u(ρ)∼B (regular solution) corresponds to the Friedrichs self-adjoint extension, selected uniquely by the Hopf topology.We then prove vortex stability via a variational-virial argument: the second variation of the Hamiltonian is positive-definite on the physical sector, and the virial identity d²I/dt²≥0 is established for all admissible perturbations. The essential spectrum is stable and no negative eigenvalue exists for the physical extension.Finally, we show that the oscillation modes of macroscopic PEW vortices — tilt mode (zero mode of rotational symmetry), bending modes, and breathing mode — map directly onto observed astrophysical phenomena: the warping and precession of galactic discs, the vertical oscillation of the Solar System about the galactic plane (period ∼64 Myr), and the helicoidal orbit of the Sun in the Galaxy. These constitute the hierarchical vortex dance, a family of topologically protected oscillations spanning from the proton (femtometre scale) to the observable universe, whose periods scale as α_em⁻²⁽ⁿ⁻⁹⁾ across the φⁿ harmonic hierarchy. Falsifiable predictions for warp precession periods and rotation curve anomalies are identified.