Pati–Salam structure from the bundle of Lorentzian metrics: a trace-reversal construction with gravitational motivation

We present a self-contained computation showing that the Pati–Salam gaugegroup SU(4)×SU(2)L×SU(2)R and the fermion content of one chiral generationof the Standard Model (including a right-handed neutrino) arise from the geometryof the bundle of pointwise Lorentzian metrics over a four-dimensional spacetimemanifold. The fibre metric belongs to the one-parameter DeWitt family Gλ, andsignature(6,4)—which leads to Pati–Salam via SO(6,4) and its maximal compactsubgroup—obtains for λ<−1/4. The trace-reversal value λ=−1/2 correspondsto general relativity. We further show that this choice is not ad hoc: the 3+1decomposition of Gλ identifies λ=−α, where α is the Wheeler–DeWitt parameterstudied by Giulini and Kiefer, who showed that gravity is attractive if and onlyif α>1/3. This condition forces λ<−1/3<−1/4, placing the fibre metricin the Pati–Salam regime. The observational constraint from primordial heliumabundance, 0.4<α<0.55, confirms that the entire physically allowed range yieldssignature(6,4). No fields beyond the spacetime metric are introduced. We statewhat this result does and does not establish regarding physical unification.