Gravitational-Wave Propagation in Quadratic Einstein--Cartan--Dirac Gravity: A Conditional Tensor-Sector Analysis
We study leading tensor propagation in a specified first-order Einstein--Cartan--Dirac theory with the parity-even quadratic torsion invariant $ζT_{λμν}T^{λμν}$. Eliminating the nondynamical Lorentz connection on the regular algebraic branch gives a metric description containing the Einstein--Hilbert term, the Dirac action, and a $ζ$-dependent axial--axial fermion contact interaction. We evaluate the transverse-traceless (TT) Hessian under an explicit factorized mean-field closure, with vanishing connected-current TT response, vanishing first- and second-order TT response of the factorized current scalar, and zero intrinsic TT anisotropic stress. Under these imposed conditions, the leading two-derivative tensor action has $Q_T=M_{\rm Pl}^{2}$ and $c_T=1$, with no leading dispersion or propagation-induced polarization splitting. These conditions do not follow from isotropy of the one-point axial current, and the present calculation does not derive them from a microscopic fermionic state. The result is therefore conditional and does not establish a general Einstein--Cartan--Dirac no-go theorem. We identify the connected-current TT response that must be calculated to extend the result beyond this closure. Published NANOGrav summaries are used only to check consistency with the GR-like prediction; no new timing-data likelihood or constraint on $ζ$ is obtained.