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 = 2Vη\,(V² r² + 9η² - 36)/9r³, establishing that non-zero Pontryagin number arises from torsion mixing Vη, not from the twist field itself. Second, we introduce the full shear tensor as the traceless symmetric tensor hᵢⱼ and organise its five components as the spin-2 quintet qA. At the isotropic point Schur's lemma forces the spin-2 Hessian to HAB = μq²\,δAB, μq² = 48π², a 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-ε side at εc+=0.48324, while on the negative side the m=± 2\ sector undergoes a second-order transition into the s≠ 0 phase at εc-=-0.29309± 0.00019. Third, in the spin-1 sector the Hessian and field-space metric of the raw variables (ωᵢ,δᵢ) are analysed as the generalised eigenvalue problem Hv = λ G v. The field-space metric is rank-deficient; the null direction Yₖ ∝ \!(1,2L/r₀) is neither gauge nor variable redundancy, but a non-dynamical auxiliary direction. The physical spin-1 mode is Xₖ = -2ωₖ + 3δₖ√13, The finite generalised eigenvalue obtained after Schur elimination of the second-class auxiliary direction is λphys -0.4183<0, with positive kinetic norm. Thus the physical spin-1 triplet is tachyonic but non-ghost. A mixed-Hessian computation confirms that the corrected spin-1 mode 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.