Dirac Spinor Emergence from First-Order Factorization in SU(2)-Covariant Time–Scalar Spectral Geometry

We establish a structural route from the TSFT covariant scale-chain operator to a Dirac-type first-order dynamics. Building on the self-adjoint SU(2)-covariant spectral geometry previously constructed for the TSFT scale chain, we show that a first-order square-root operator acting on a minimal doubled module exists and that the algebra required to eliminate cross terms forces an emergent Clifford (Pauli) structure. The resulting Dirac-form operator squares to a purely quadratic generator comprising a scale-direction contribution and an internal holonomycontrolled gap term depending only on the SU(2) conjugacy invariant Tr(W) = 2 cos α. In a translation-invariant baseline, we derive an exact dispersion bridge relating the symmetric first-order generator to the TSFT Laplacian symbol, yielding a controlled lattice correction. In the low-spectrum regime this Dirac construction reduces to a Schr¨odinger-type evolution generated by the TSFT second-order operator, providing a consistent nonrelativistic limit. The contribution is structural and operator-theoretic: no Standard Model fermions are identified and no Higgs/Yukawa mechanisms are invoked.