Classification of Pati--Salam Asymmetric $\mathbb{Z}_2 \times \mathbb{Z}_2$ Heterotic String Orbifolds
We develop a systematic classification of asymmetric $\mathbb{Z}_2$ orbifold actions in Pati-Salam heterotic string vacua constructed in the free fermionic formulation. Starting from symmetric $\mathbb{Z}_{2} \times \mathbb{Z}_{2}$ orbifold vacua with an $SO(10)$ GUT, we allow the Pati--Salam breaking vector to act asymmetrically on the internal degrees of freedom. The asymmetric orbifold action freezes geometrical moduli whilst inducing doublet--triplet splitting in the untwisted sector. Notably, this doublet-triplet splitting operates for any asymmetric action, including pure asymmetric shifts that preserve all geometric moduli, and is therefore independent of moduli stabilisation. Classifying the breaking vector according to its twist action, we find six inequivalent classes of geometric moduli spaces characterised by 12, 8, 4 or 0 real untwisted moduli. Through combining these asymmetric twists with all compatible asymmetric shifts, 24 inequivalent cases are identified and characterised by their residual moduli content and internal Narain lattice. For each case we construct representative basis sets admitting three chiral generations, providing the starting point for further classification within each class. We perform explicit GGSO (generalized Gliozzi-Scherk-Olive) phase enumerations in representative model classes with 12, 8, 4 and 0 moduli, classify the resulting $\mathcal{N}=1$ and $\mathcal{N}=0$ vacua according to phenomenological criteria and identify exophobic, phenomenologically viable models. We compute the partition function and corresponding one-loop vacuum energy at the free fermionic point in moduli space for each phenomenologically viable model across the four classes. As the number of geometrical moduli decreases, the number of distinct partition functions for these vacua collapses to a small number, reflecting a pronounced degeneracy under GGSO phase variations.