Chromatic Gravitational-Wave Tests of Dirac-Induced Torsion in Quadratic Einstein--Cartan Gravity: Nanohertz PTA Null Signatures

We derive a conditional result for leading tensor propagation in Einstein–Cartan–Dirac gravity with the parity-even invariant $\zeta T_{\lambda \mu \nu }{T}^{\lambda \mu \nu }$ and algebraic torsion sourced by a coarse-grained fermionic axial current. On a spatially flat, homogeneous, and isotropic background, eliminating the nondynamical Lorentz connection yields the Einstein–Hilbert transverse-traceless (TT) action at leading two-derivative order if the matter state obeys an explicit factorized mean-field closure: the connected contribution to $⟨J_{5\mu }{J}_5^{\mu }⟩$ is set to zero, the factorized scalar ${\bar{J}}_{5\mu }{\bar{J}}_5^{\mu }$ has vanishing first- and second-order TT response, and intrinsic TT anisotropic stress vanishes. Under these conditions, $Q_T=M_{\mathrm{P}\mathrm{l}}^2$, the tensor speed is luminal, and the local dispersion relation is ${\omega }^2=k_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}^2$, with no leading propagation-induced polarization splitting. The contribution is a model-specific constrained TT-Hessian calculation for this invariant that retains the auxiliary-connection response and states the closure needed for the $\zeta$-dependent terms to cancel. The connected-current TT response of a general fermionic state is not evaluated; this is therefore a conditional result, not a general Einstein–Cartan no-go theorem. The coupling $\zeta$ remains in the axial–axial fermion contact interaction. We define null-test observables and a covariance-weighted statistic for future pulsar-timing-array (PTA) likelihood analyses. Published NANOGrav 15-year harmonic posteriors and the public 2026 collaboration preprint serve only as consistency checks: we perform no new time-of-arrival (TOA)-level likelihood or evidence calculation, and these data do not directly constrain $\zeta$.