From Astrophysical Jets to the Origin of Spin: Axial-Vortex Solutions in Spectral Geometry as a Bridge between Scales

This work presents a unified geometric framework that links the phenomenology of astrophysical compact objects and relativistic jets to the fundamental quantum numbers of elementary particles. Starting from the identification of Yang–Mills gauge fields with spacetime contorsion within the formalism of noncommutative spectral geometry, we derive a new class of axial-vortex solutions – topological solitons breaking spherical symmetry. These solutions provide a geometric engine forrelativistic jets, operating independently of accretion, explaining observed “disk-less” active galactic nuclei, magnetar flares, and the ionization of “dark” galaxies.We demonstrate that the vortex’s helical magnetic field naturally collimates and accelerates particles via a geometric electromotive force, with predicted jet powers consistent with observations.Delving deeper, we show that the dynamics of field perturbations in this back- ground obey a generalized geodesic equation, erasing the distinction between ge-ometry and matter propagation. Furthermore, the vortex acts as a chiral filter, generating observable circular polarization patterns. Most fundamentally, we prove that the spin-statistics connection of quantum fields emerges from the topology of the underlying spectral triple: bosonic degrees of freedom are associated with spherical topology (Sn) and characteristic classes, while fermionic degrees of freedom are inherently toric (T n), governed by spin structures and modular parameters. The axial-vortex physically incarnates this correspondence, with its monopole charge (sphere) and fermionic zero modes (torus). This work thus establishes a concretepathway from the macroscopic dynamics of cosmic jets to the geometric origin of spin and statistics, offering testable predictions across astrophysics and particle phenomenology.