Lorentzian Einstein-Cartan Minisuperspace with Nieh-Yan Torsion: Hamiltonian Branches, Pontryagin Diagnostics, and Weyl-Source Obstructions

We reduce Einstein--Cartan gravity with a Nieh--Yan term to a Lorentzian closed-FLRW S³ minisuperspace and formulate the α=0 EC+NY branch as the principal Hamiltonian-admissible branch. Retaining the lapse $N(t)$ through variation and treating the EH bulk action, GHY-type boundary term, and non-propagating torsion variables η(t) and $V(t)$ as algebraic constraints in the same convention, we show that the closed-FLRW Hamiltonian constraint and the torsion auxiliary sector close consistently. We then separate the form-Hodge Pontryagin density Pform from the internal-pair diagnostic Pᵢₙₜ. In the Lorentzian reduced branch, Pform is exactly zero, while in the MX sector Pᵢₙₜ can be non-zero as a separate internal channel. This non-vanishing does not violate Hamiltonian admissibility. For the Weyl² coupling branch (α≠0), the Levi-Civita Weyl-source branch is conformally silent on isotropic FLRW but exhibits a lapse-sensitive spin-2 Weyl obstruction on a homogeneous spin-2 local jet. We further show that for the EC Weyl-source branch, none of the AX, VT, or MX torsion modes cancels this Weyl witness locally, in the sense of removing both the spin-2 acceleration Hessian witness and the lapse-sensitive Ṅ witness within the tested homogeneous spin-2 local-jet class. This constitutes a local non-cancellation criterion and a sufficient local diagnostic separating the α=0 baseline from the α≠0 Weyl-source branches.