The Origin of Spin and Planck's Constant: Axial-Vortex Solutions in Spectral Geometry as a Bridge between Quantum Geometry and Cosmology

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 and the very nature of quantization itself. 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 for relativistic jets, operating independently of accretion, explaining observed “diskless” 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 background obey a generalized geodesic equation, erasing the distinction between geometry 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. Theaxial-vortex physically incarnates this correspondence, with its monopole charge(sphere) and fermionic zero modes (torus).The culmination of this work is the demonstration that Planck’s constant ℏ, the quantum of action, is not an independent postulate but a derived scale. We showthat ℏ emerges as the conversion factor between the dimensionless spectral action of noncommutative geometry and the empirical action of physics, fundamentally tied to the noncommutativity scale ΛNC. This geometric origin of quantizationprovides a natural resolution to the measurement problem via a predicted stochastic gravitational wave background and establishes a deep link between the cosmological constant, unification scale, and the value of ℏ. This work thus establishes a concretepathway from the macroscopic dynamics of cosmic jets to the geometric origin of spin, statistics, and quantization, offering testable predictions across astrophysics, particle phenomenology, and foundational quantum mechanics.