Reduced-Sector $\chi$-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 $S^3$, $T^3$, $Nil^3$, and $Sol^3$, restricted to the $\alpha=0$ EC+NY reduced branch. For $Nil^3$ and $Sol^3$ 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 $P_{form}$, the internal-pair diagnostic $P_{int}$, and the Nieh--Yan endpoint channel $Q_{NY}$. The form-Hodge density $P_{form}$ vanishes exactly on the active torsion branches by block orthogonality. The diagnostic $P_{int}$ on the MX branch carries the topology dependence in a single coefficient, $C_{top} = -9\chi.$ Furthermore, the reduced vacuum orbit atlas on the auxiliary shell reduces to $\dot q^{2} + \chi = 0.$ Consequently, $\chi>0$ admits no real initial data on the reduced vacuum atlas, $\chi=0$ gives static or degenerate behaviour, and $\chi<0$ gives monotonic expansion together with singular-approach collapse sheets. The two relations $C_{top}=-9\chi$ and $\dot q^{2}+\chi=0$ are obtained from independent computational routes and converge on the same scalar $\chi$; this convergence is the content of the $\chi$-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.