Curvature Identities and Restricted Potential Bounds in Einstein--Cartan--Nieh--Yan Minisuperspace with Irreducible EC-Weyl Corrections

We evaluate the Einstein--Cartan (EC) action with a constant complete Nieh--Yan (NY) coupling and an irreducible EC-Weyl-squared term on static homogeneous families of $S^3\times S^1$, $T^3\times S^1$, and a fixed compact $\mathrm{Nil}^3\times S^1$. Varying radius and volume-preserving shape at fixed torsion parameters and circle circumference, we classify boundedness and attainment of minima of the reduced potential. The complete NY density and the form-index Pontryagin density vanish for the prescribed connection, while the Weyl contribution is governed by the spatial shape dependence. We prove that the irreducible EC-Weyl tensor equals the Levi-Civita (LC) Weyl tensor throughout these families, so their reduced potentials and classifications coincide. A brief comparison isolates the additional curvature sectors in a different Ricci-tracefree quadratic action. For $S^3$ at nonpositive Weyl coupling, an exact factorization of the shape difference reduces global minimization in two variables to the round slice. A unique finite-radius minimum exists for $V\ne0$ and $\eta^2