Scale-Dependent Unification of Statistics and Dynamics: Emergence of the Dirac Field from a Scalar Progenitor (Part IV)

We present a unified framework where the foundational structures of the Standard Model—Dirac like spinors, Abelian gauge interactions, and Fermi statistics—emerge not as independent postulates, but as necessary geometric consequences of a single higher-derivative scalar Lagrangian. Building on our preceding work regarding ultraviolet uniqueness (Part I), algebraic onsistency (Part II), and the emergent Born rule (Part III), we utilize the six-component Feshbach-Villars formalism as a bridge between the Lee-Wick scale (M ≈ 11.3 TeV) and low-energy physics. By analyzing this dynamics within a Split-Quaternionic Krein space, we derive the non-relativistic limit via a generalized Foldy-Wouthuysen transformation, reproducing the Darwin term with its precise coefficient (1/8m2). Furthermore, we identify the gauge field as an emergent Berry connection and demonstrate that the Pauli exclusion principle originates from the topological π-Berry phase inherent to the underlying geometry. Crucially, this framework implies a scale-dependent transmutation of statistics: the system behaves as a fermion in the infrared effective limit, while asymptotically restoring its fundamental bosonic scalar nature in the ultraviolet limit, suggesting a radical unification of matter and force.