Quaternionic superconductivity links spinful pairing, topology, and charge- 4e order

We present a unified theoretical description of spinful superconductivity that combines singlet and triplet pairing within a complexified quaternion order parameter. The formulation provides a compact route from the superconducting gap to the Bogoliubov-de Gennes spectrum and its symmetry constraints. For a spin-independent normal state, mixed singlet-triplet pairing generally produces two gap branches, while the conventional degenerate spectrum is recovered for unitary pairing. We extend the framework to a scalar composite charge- 4e field and derive its fluctuation-induced mass correction with an explicit field normalization. The resulting Ginzburg-Landau theory yields the conditions for a local vestigial instability, phase locking, and h/4e flux quantization. A Rashba-lattice example illustrates the distinction between class-DIII symmetry and nontrivial topology. Finite-size density matrix renormalization group data show slower quartet than pair decay over the accessible distance range, with residual bond-dimension dependence. Illustrative vortex and Josephson responses connect the field theory to experimentally accessible signatures while distinguishing coherent two-pair transport from evidence for a bulk charge- 4e condensate.