A Geometric Derivation of Planck's Law and the Fine-Structure Constant from Solid-Torus Vacuum Topology

We present a purely geometric derivation of blackbody spectral radiance and the fine-structure constant αfs from the topology and harmonic structure of a vacuum flow 1-form defined on a solid-torus configuration space. No quantum postulates are assumed: discrete energy levels arise solely from topological winding constraints on the noncontractible S1 cycle. The geometric action constant αgeom, derived from circulation quantisation of harmonic 1-forms, plays the role of Planck’s constant. When the electromagnetic circulation invariant is included, the physical fine-structure constant α_fs = e^2/(4πε0cℏ) emerges as a dimensionless ratio of two geometric invariants: a toroidal action unit and an electromagnetic flux unit. This demonstrates that Planck’s constant and α_fs are not fundamental but arise as shadows of deeper geometric quantities.