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 α C²EC 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 ωEC = ΓLC + 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, C²EC = C²LC holds exactly in the AX ($V=0$) and VT (η=0) backgrounds; the EC correction appears only in the MX background as C²EC = C²LC + 16V²η²/(3r²). Theorem 2 (Palatini protection, EC): GECηη = GECVV = 0 holds algebraically; torsion remains non-propagating and the spin-2 kinetic structure is unchanged from the LC case. Theorem 3 (δ-sector null theorem): The auxiliary cubic coefficient C_δ := ∂³ Veff∂δ₀∂δ₁ ∂δ₂|δ=0 vanishes exactly in all three topologies and all torsion backgrounds, and ∂_α C_δ = 0. Theorem 4 (Palatini universality): In S³ × S¹ the AX and VT spin-2/spin-1 mass spectra are identical: spin-2 mass μ² = 128π² Lr/(3κ²) - 16384π² Lα/(3r) (five-fold degenerate) and spin-1 mass m²_ω = 16π² L³/(κ² r) (three-fold degenerate). Theorem 5 (torsion--volume coupling): In T³ × S¹, ∂² Veff/∂ε∂ s|₀ = 48π⁴ Lη²r/κ² ≠ 0; the cross-term is of purely kinematic origin, independent of the Weyl mass. Theorem 6 (Nil³ EC slice minimum): For α<0, Nil³ × S¹ supports a positive EC-induced slice minimum at r₀ = (4κ/\!√3)√|α| on the η=V=0 section; the full homogeneous Hessian is positive definite for |κ²θNY|<1, marginal at equality, and a saddle above it. The scaling exponent is γ=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(ω₂)≠ 0 (non-commutative direction) while m²(ω₀)=m²(ω₁)=0 (commutative directions). Theorem 9 (ε-s cross-term classification): The origin of ∂² Veff/∂ε∂ 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.