Fermionic Geometry: The Period-Doubling Origin of Half-Integer Spin in Topological Time Crystals

The Standard Model establishes fundamental fermion spin through the representation theory of the Poincaré group and attributes rest mass to the dynamical Higgs mechanism. Establishing a geometric ontology for these properties requires mapping point-like singularities to extended structures within a topological quantum field theory. A Faddeev-Skyrme soliton framework is derived by integrating out the heavy Higgs radial mode utilizing the Schwinger-Keldysh closed-time-path formalism. This non-equilibrium quantum field theory approach accurately captures the quantum vacuum back-reaction, replacing classical stochastic phenomenological models and anchoring the topological rigidity coefficients directly to the electroweak vacuum expectation value. The resulting macroscopic topological soliton exhibits spontaneous breaking of discrete time-translation symmetry under external driving. The topological classification of the Q=1 Hopfion configuration space imposes absolute geometric boundaries on the system's parameter space evolution. The application of the Finkelstein-Rubinstein topological constraint dictates that a complete temporal drive cycle generates a precisely quantized Berry phase of π. This geometric phase rigidly locks the single-period Floquet evolution operator to an eigenvalue of -1, enforcing a deterministic macroscopic 2T period-doubling dynamics. This exact sub-harmonic rigidity provides a direct mathematical isomorphism to the 4π rotational symmetry characteristic of half-integer spinors. The rest mass of the emergent fermionic state is defined by the discrete spectrum of the Dirac operator evaluated on the curved background manifold of the Hopfion. The geometric excitation spectrum predicts a macroscopic neutral topological state, T^0, at the 2.43 GeV energy scale. The physical observability of this topological state is fundamentally restricted by an exponential instanton tunneling barrier in zero-temperature vacuum environments. Quantitative cross-section calculations demonstrate that sphaleron transitions within the Quark-Gluon Plasma of heavy-ion collisions neutralize this topological suppression. This thermal transition mechanism provides a direct, measurable production rate, establishing a strict falsifiability criterion for experimental verification at the ALICE detector facility.