Reduced-Sector χ-Universality in Lorentzian EC+NY Minisuperspace: Topology-Robust Admissibility, P-Channel Diagnostics, and Reduced Vacuum Orbits

We analyse the topology dependence of the Lorentzian Einstein--Cartan plus Nieh--Yan (EC+NY) reduced sector across the four homogeneous three-geometries S³, T³, Nil³, and Sol³, restricted to the α=0 EC+NY reduced branch. For Nil³ and Sol³ the treatment is confined to the isotropic-scale reduction. In this reduced sector the Hamiltonian constraint and the torsion auxiliary equations exhibit a topology-robust admissibility pattern: the EH, AX, and VT branches are admissible, while the MX branch is conditionally admissible through a real auxiliary branch. For Pontryagin-type quantities we separate the form-Hodge density Pform, the internal-pair diagnostic Pᵢₙₜ, and the Nieh--Yan endpoint channel QNY. The form-Hodge density Pform vanishes exactly on the active torsion branches by block orthogonality. The diagnostic Pᵢₙₜ on the MX branch carries the topology dependence in a single coefficient, Cₜₒₚ = -9χ. Furthermore, the reduced vacuum orbit atlas on the auxiliary shell reduces to q² + χ = 0. Consequently, χ>0 admits no real initial data on the reduced vacuum atlas, χ=0 gives static or degenerate behaviour, and χ<0 gives monotonic expansion together with singular-approach collapse sheets. The two relations Cₜₒₚ=-9χ and q²+χ=0 are obtained from independent computational routes and converge on the same scalar χ; this convergence is the content of the χ-controlled topology universality. The results are a classification within the reduced Lorentzian EC+NY sector; global, anisotropic, and matter-coupled extensions are left to future work.