Homogeneous Three–Topology Comparison and Mode Dictionary in Einstein–Cartan + Nieh–Yan Theory:Geometric Structure of EC–Weyl Coupling

We systematically analyse the homogeneous minisuperspace of gravity with Einstein--Cartan (EC) connection and Weyl-squared coupling C²₄₂ across three spatial topologies---S³ S¹, T³ S¹, and Nil³ S¹---and determine how the EC--Weyl coupling modifies the mode dictionary of each topology. Extending the Levi--Civita framework of papers~I--III to the EC connection ₄₂ = ₋₂ + K (with K the contortion), we classify torsion modes as axial-scalar (AX), vector-trace (VT), and their product (MX), and establish nine theorems. Theorem 1 (AX/VT dropout): In all three topologies, C²₄₂ = C²₋₂ holds exactly in the AX (V=0) and VT (=0) backgrounds; the EC correction appears only in the MX background as C²₄₂ = C²₋₂ + 16V²²/ (3r²). Theorem 2 (Palatini protection, EC): G^EC_ = G^ECₕₕ = 0 holds algebraically; torsion remains non-propagating and the spin-2 kinetic structure is unchanged from the LC case. Theorem 3 (-sector null theorem): The auxiliary cubic coefficient C_: = ³ V₄₅₅₀₁ ₂|=₀ vanishes exactly in all three topologies and all torsion backgrounds, and _ C_ = 0. Theorem 4 (Palatini universality): In S³ S¹ the AX and VT spin-2/spin-1 mass spectra are identical: spin-2 mass ² = 128² Lr/ (3²) - 16384² L/ (3r) (five-fold degenerate) and spin-1 mass m²_ = 16² L³/ (² r) (three-fold degenerate). Theorem 5 (torsion--volume coupling): In T³ S¹, ² V₄₅₅/ s|₀ = 48⁴ L²r/² 0; the cross-term is of purely kinematic origin, independent of the Weyl mass. Theorem 6 (Nil³ EC slice minimum): For <0, Nil³ S¹ supports a positive EC-induced slice minimum at r₀ = (4/\!3) || on the =V=0 section; the full homogeneous Hessian is positive definite for |²₍ₘ|<1, marginal at equality, and a saddle above it. The scaling exponent is =1/2. Theorem 7 (Nil³ quintet splitting): The spin-2 quintet around the EC slice-minimum branch splits as 5 0+2+1+1; the zero mode q₃ is geometrically protected by the one-dimensional Lie structure of Nil³. Theorem 8 (uniaxial mass splitting): In the Nil³ spin-1 sector, m² (₂) 0 (non-commutative direction) while m² (₀) =m² (₁) =0 (commutative directions). Theorem 9 (-s cross-term classification): The origin of ² V₄₅₅/ s differs essentially across topologies: purely kinematic for T³, vanishing for S³, and of pure curvature/Weyl origin for Nil³. These results establish a closed comparison framework for the homogeneous bulk geometric sector in EC+NY theory under the axial+vector-trace torsion ansatz.