Unified Geometric Landau EFT of Homogeneous S³ S¹ Minisuperspace in Einstein-Cartan + Nieh-Yan Theory

We reconstruct the homogeneous S³ S¹ minisuperspace reduction of Einstein-Cartan (EC) gravity augmented by the Nieh-Yan (NY) term, and demonstrate that the homogeneous geometric sector admits a unified description as a finite-degree-of-freedom geometric Landau effective theory. The aim is to organise the Pontryagin protection structure, the spin-2 shear deformations, and the spin-1 twist-mixing system along the irreducible spin-0/2/1 modes, thereby reconsolidating them into a single theoretical framework. First, we show that the Pontryagin density P= R, R vanishes identically under the AX and VT torsion modes, while only the MX mode can sustain P 0. In the MX mode the source at the isotropic point is\ P₀ = 2V\, (V² r² + 9² - 36) 9r³, that non-zero Pontryagin number arises from torsion mixing V, not from the twist field itself. Second, we introduce the full shear tensor as the traceless symmetric tensor h₈₉ and organise its five components as the spin-2 quintet qA. At the isotropic point Schur's lemma forces the spin-2 Hessian to\ H₀₁ = q²\, ₀₁, q² = 48², five-fold degeneracy. Under a squash background the SO (3) SO (2) splitting reveals that the m=0 singlet is the first soft mode on the positive- side at ₂+=0. 48324, while on the negative side the m= 2 sector undergoes a second-order transition into the s 0 phase at ₂-=-0. 29309 0. 00019. Third, in the spin-1 sector the Hessian and field-space metric of the raw variables (ᵢ, ᵢ) are analysed as the generalised eigenvalue problem\ Hv = G v. field-space metric is rank-deficient; the null direction\ Yₖ \! (1, 2Lr₀) neither gauge nor variable redundancy, but a non-dynamical auxiliary direction. The physical spin-1 mode is\ Xₖ = -2ₖ + 3ₖ13, generalised eigenvalue ₏₇ₘₒ = 4. 1867 > 0. A mixed-Hessian computation confirms that Xₖ fully decouples from the spin-2, radial, and torsion sectors at quadratic order. Taking all results together, the model is understood as a unified geometric Landau-EFT on the homogeneous S³ S¹ geometry, in which the homogeneous geometric degrees of freedom are organised along irreducible representations and the Pontryagin protection, phase-transition directions, physical/auxiliary decomposition, and allowed interaction terms are integrated into a single theoretical framework. This work provides one closed framework for the classical minisuperspace geometric sector in EC+NY theory and serves as a foundation for future extensions including quantisation.