A Certified Torsion-Free G₂ Structure on a TCS Neck Model via Computer-Assisted Proof

We construct an explicit G₂ structure on the TCS-type neck model K3 × T² × [0,1] and certify, via Newton–Kantorovich interval arithmetic with zero finite differences, the existence of a unique nearby torsion-free G₂ structure with contraction parameter h ≤ 8.95 × 10⁻⁹ (analytical certificate, β = 0.321 from a Sturm–Liouville bound), sharpened to h ≤ 1.43 × 10⁻⁹ with the numerical β = 0.02961; both well below the threshold 1/2 and relative to the fixed K3 neural-network input. The solution space is finite-dimensional (ℝ¹⁶⁸), the torsion map is a polynomial of degree ≤ 10 in the neck coordinate, and the analytical inverse bound is derived from the algebraically exact determinant det(g) = 65/32. The optimized 169 parameters (168 Chebyshev–Cholesky coefficients + ACyl decay rate γ) exhibit a five-constant approximate structure (g_ss ≈ 19/6, g_{T²} ≈ 7/6, g_{K3} ≈ 64/77, det(g) = 65/32, γ = 2π√(6/7)) with sub-percent deviations; the first two are torsion minimizers and the last is derived from the T² Hodge Laplacian and H¹(K3) = 0. The G₂ torsion-free condition acts as an eigenvalue equalizer, reducing the K3 fiber anisotropy from 10:1 to 1.012:1. We prove analytically that product-type Ricci-flat G₂ metrics satisfy spectral democracy (Theorem 1.3). The NK certificate extends to the full 7D product metric via an analytical Fréchet bound on the K3 fiber, giving ‖T(g*)‖_{C⁰} ≤ 1.59 × 10⁻³, well below the Joyce perturbation threshold ε₀ ≈ 0.1. The underlying seam-sector torsion at the NK iterate is η ≤ 2.949 × 10⁻⁵ (direct sup of √(|dφ|² + |d⋆φ|²) at Clenshaw–Curtis nodes, used as η in NK; the slightly larger 2.984 × 10⁻⁵ in Table 8 is the conservative √((sup|dφ|)² + (sup|d⋆φ|)²)), and the empirical 7D maximum measured directly over 220,000 K3 fiber points is 3.154 × 10⁻⁵. Conditional on the existence of a compact simply connected 7-manifold with Betti numbers (b₂, b₃) = (21, 77) (a pair absent from all known G₂ constructions) this certificate yields a torsion-free G₂ metric on the compact manifold via Joyce's perturbation theorem. Keywords: G₂ holonomy, computer-assisted proof, Newton-Kantorovich, interval arithmetic, Chebyshev-Cholesky metric, spectral democracy