We analyse the Lorentzian first-order Einstein--Cartan plus Nieh--Yan (EC+NY) system on Bianchi class-A backgrounds using a Dirac-before-Routh (DBR) constraint-reduction protocol. The rank defect of the Bianchi class-A structure matrix is a mathematically known invariant; our result is that this defect is realised, in the DBR-standard quadratic constrained perturbation spectrum, as a residual non-propagating scalar kernel. Across the four backgrounds T³, Nil³, Sol³, and S³, the propagating tensor sector contains two positive-kinetic transverse-traceless (TT) graviton degrees of freedom, and the dimension of the scalar kernel is dₙ₄ₑ₎ = n = 3-rank (n). This classification is established through the DBR constraint algebra, a frame-invariance check, scalar/tensor spectral analysis, and a scoped k>0 spectral completion over the analysed harmonic sectors. Reversing the order of operations -- eliminating the auxiliary variables before Dirac closure -- produces a Dirac-inequivalent extra scalar mode in the verified S³ scalar sector; in light of this control result, all subsequent physical claims are made within the DBR protocol. We further classify the finite deformations of the zero kernel into exact moduli and curvature walls within a finite Palatini branch, and we show that the torsion ansatz scalars remain an exact auxiliary under the NY-MAIN route, acquiring no kinetic or gradient terms even under field promotion, so that the torsion sector does not affect the above hierarchy of results. All results are established within the explicitly stated quadratic, harmonic-sector, and homogeneous scopes.
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