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

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.The stability analysis presented here is valid in the plane of the operculum (2D cross-section of the vortex shell). Out-of-plane stability of operculum oscillation modes is addressed in Article 7.