Pati-Salam gauge unification from the Lorentzian DeWitt metric

We show that the DeWitt supermetric on the space of Lorentzian metrics over a four-dimensional manifold has signature (6,4), in contrast to the well-known Euclidean signature (9,1). The resulting structure group SO ( 6 ) × SO ( 4 ) ≅ SU ( 4 ) × SU ( 2 ) L × SU ( 2 ) R is the Pati-Salam gauge group. Equal Dynkin indices give sin 2 θ W = 3 / 8 at unification, running to 0.2312 at M Z . Clifford analysis of the positive-norm bundle yields one complete fermion generation as a 16 -plet of Spin ( 10 ) ⊃ SO ( 6 , 4 ) . A spin c index theorem on a K3 base gives exactly three generations. The Gauss equation for the metric embedding produces the Einstein–Hilbert, Yang–Mills, and scalar terms with correct relative signs. All eight results follow from the single geometric datum of the DeWitt metric evaluated on a Lorentzian background, with no free parameters in the gauge or generation sectors.