Fermion Representations from the Metric Bundle: Clifford Algebras and the Pati-Salam Spinor

Paper 2 of 6 in the Metric Bundle Programme. In a companion paper, we showed that the DeWitt supermetric on the space of Lorentzian metrics over four-dimensional spacetime has signature (6,4), yielding the Pati-Salam gauge group (4) × (2)_L × (2)_R as the maximal compact subgroup of (6,4). In this paper, we derive the fermion content of the theory from the Clifford algebra of the positive-norm sector of the normal bundle. The Clifford algebra (^6) ⊗ ≅ M_8() acts on an eight-dimensional spinor ^8, which decomposes under the chirality operator as ^8 = S^+ ⊕ S^- = 4 ⊕ 4̄ of (4) ≅ (6). We prove that the (3) subalgebra defined as the centraliser of a complex structure J in so(6) coincides exactly with the (3) from the Pati-Salam breaking (4) → (3) × (1)B-L, with the (1) charge reproducing the baryon-minus-lepton number in the ratio +1/3 : -1 (triplet : singlet). Combined with the (2,1) ⊕ (1,2) spinor of (4) ≅ (2)_L × (2)_R, the full representation is the 16 of (10) ⊃ (6) × (4): one complete generation of Standard Model fermions including a right-handed neutrino, with all hypercharges correctly reproduced by Y = (B{-}L)/2 + T3R. Part of a six-paper series deriving the Pati-Salam gauge group, fermion content, gauge dynamics, anomaly cancellation, and the three-generation structure from the geometry of the metric bundle Y14 = Met(X4). v2 changes (March 2026): Added {slashed} for correct rendering of Dirac operator notation. v3 changes (March 2026): v3: Updated Discussion with forward references to Papers IV-VI (anomaly cancellation verified, three-generation conjecture, Yukawa structure). Added Paper4/Paper5/Paper6 bib entries.