Spectral Geometry of an Explicit G₂ Metric: Laplacian Spectrum and Harmonic Forms

We compute the Laplacian spectrum and harmonic forms on the certified torsion-free G₂ structure of a companion paper [A], constructed on a TCS-type neck model K3 × T² × [0,1]. The scalar spectral gap is λ₁ = 0.12461 ± 0.00016 (Richardson-extrapolated across six grids, Appendix A) with Weyl exponent α = 1.998 (adiabatic prediction: 2.0), matching the analytical formula λ₁ = π²/(L² g_ss) = 6π²/475 ≈ 0.12467 (derived a posteriori from the NK-certified seam parameters) to 0.05%. Spectral analysis of the Hodge Laplacians realizes the harmonic multiplicities (b₂, b₃) = (21, 77) expected from the conjectural compact extension: 21 near-zero eigenvalues of Δ₂ (gap ratio 14,635) and 77 near-zero eigenvalues of Δ₃. The 21 harmonic 2-forms carry a K3 fiber intersection form of signature (3, 18), rank-21 restriction of the ambient K3 lattice (3, 19), with SD/ASD eigenvalue gap 2,210. The spectral democracy theorem proven in [A] is verified numerically to 10⁻⁵ precision. The complete Kaluza-Klein tower comprises 22,671 levels with Poisson level spacing statistics. All spectral results are properties of the explicit neck-model metric; their interpretation on a closed compact G₂ manifold is conditional on the existence of such a manifold with (b₂, b₃) = (21, 77). Keywords: spectral geometry, G₂ holonomy, Hodge Laplacian, Kaluza-Klein spectrum, harmonic forms, intersection form