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 solutionsprovide 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 fieldnaturally 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). 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 show that 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 quantization provides a natural resolution to the measurement problem via a predicted stochasticgravitational wave background and establishes a deep link between the cosmological constant, unification scale, and the value of ℏ. Building on this geometric foundation, we prove three theorems that elevate the framework from a reformulation to a first-principles derivation of quantum mechanics itself. First, the KO-dimension of the spectral triple is forced to be 6 — not by fiat, but by the stability of the axial-vortex and the observed Nf = 3 fermion generations. Second, the canonical anticommutation relations (CAR) of fermionic fields are shown to be topologically mandatory: the configuration space of the Dirac operator is homotopy equivalent to Map(T 2, U(Nf)), whose non-trivial fundamental group π1∼= Z forces a sign-changing Pfaffian in the path integral, making commutation relations inconsistent. Third, Planck’s constant ℏ is derived exactly as ℏ = g2ϕ40|θ|2/2π from the topological quantization of the Wilson loop in the noncommutative vortex core — no free parameters, no empirical adjustment. The noncommutativity scale is thus fixed bythe electroweak scale, not by the Planck scale; the latter emerges only when ϕ0 is elevated to MPl. The axial-vortex thereby serves as an *experimentum crucis*: it is simultaneously a topological soliton observable by telescopes and the geometric origin of the quantum of action itself. This work thus establishes a concrete pathway from the macroscopic dynamics of cosmic jets to the geometric origin ofspin, statistics, and quantization, offering testable predictions across astrophysics, particle phenomenology, and foundational quantum mechanics.