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

We recast spinful superconductivity as a \textit{quaternion field theory}, where a quaternion is a four-component hypercomplex number with units $(\boldsymbol{e}_x,\boldsymbol{e}_y,\boldsymbol{e}_z)$, that encodes the spin-singlet/triplet gap in a single field $q(\mathbf{k})$. This yields a compact Bogoliubov-de Gennes (BdG) Hamiltonian $H_{\rm BdG}=\xi_{\mathbf{k}}\tau_z+\tau_+q+\tau_-\,q^\ddagger$ and keeps time-reversal symmetry, Altland-Zirnbauer classification, and topological diagnostics in the same variables. In general the mixed singlet--triplet spectrum is branch-split, while the familiar perfect-square form is recovered only for unitary pairing. We introduce a quarteting field $Q\!\propto\!\mathrm{Sc}(q^2)$ and a minimal Ginzburg-Landau (GL) functional with covariant derivatives $(\nabla-2ie\mathbf A)q$ and $(\nabla-4ie\mathbf A)Q$. Analytically, a one-loop evaluation of the fluctuation bubble $\Pi(0)$ gives a quantitative vestigial charge-$4e$ criterion $\mu_{\rm eff}=\mu-\frac{g^2}{2}\Pi(0)<0$. Numerically, we verified: (i) a two-dimensional (2D) class-DIII lattice model whose $\mathbb{Z}_2$ index, computed from the occupied BdG eigenvectors via the standard sewing-matrix (Pfaffian) construction at time-reversal-invariant momenta, matches helical edge spectra; (ii) a GL simulation of a pure-$Q$ vortex carrying $h/4e$ flux within $\sim2\%$ and exhibiting $\xi_Q\!\propto\!\sqrt{\eta/|\mu_{\rm eff}|}$; and (iii) a short-junction current-phase relation with a controlled window where the second harmonic dominates ($I_2\!\gg\!I_1$), together with doubled alternating-current Josephson emission and a Shapiro response consistent with $4e$-dominated transport. The framework provides a compact, symmetry-faithful route from microscopic pairing to device-level charge-$4e$ signatures.