Vacuum Spin-Torsion Energy Density and Two-Modulus Stability on $S^2\times\mathbb{C}P^2$

Abstract

In Einstein--Cartan--Kaluza--Klein compactification on $S^2\times\mathbb{C}P^2$, elimination of the non-propagating torsion generates a quartic spin-current contact interaction whose vacuum expectation value is nonzero even at zero fermion occupation. We reduce the ten-dimensional $\Gamma_{ABC}$ structure to the zero-mode sector and find that the reduction is channel-resolved: each of the three internal Cartan-plane bilinears contribute with the opposite sign to the external (axial) channel, so that the full trace is $R_{ch} = T_{\rm vac}/T_{\rm axial} = 1 + 3\,(-1) = -2$. The vacuum term is therefore twice the axial estimate and opposite in sign, and, combined with the elimination constant $c_{10} = -\kappa_{10}/32$, verified by explicit Clifford computation, it is attractive and proportional to $N_{\rm fam}^2$. We obtain the separated-point correlator, which in the continuum scales as $N_{\rm fam}^2 d_R/(V_6\,r^6)$, and, conditional on the physical quantum-boundary prescription, evaluate it at the minimum supported separation. The finite-density piece of the same correlator has the same sign and grows with chemical potential. Binding deepens after nucleation; the single-particle Hartree self-contraction vanishes for chiral zero modes. Using the derived $1/(a_1^2a_2^4)$ vacuum term with the Einstein-frame curvature and quantized-flux contributions, we analyze a coupled two-modulus toy potential. At fiducial couplings, the $(m,n)=(3,3)$ sector has a finite-radius stationary point with positive Hessian in both radion directions, while the four-family $(4,3)$ minimum lies beyond the adopted quantum-curvature boundary. The three-family sector $(1,5)$ is also locally stable but is less bound. 3-family dominance of the thermal nucleation weight further imposes the condition $\kappa_4 \lesssim 7.5\,(16/B)\,\ell_P^2$. These results serve as inputs to our higher-level synthesis.