Spectral Geometry of an Explicit G2 Metric on a Compact 7-Manifold

Brieuc de La Fournière (2026) Full text (markdown) | Zenodo DOI: 10.5281/zenodo.18920368

First explicit numerical computation of Kaluza-Klein spectrum on compact G2 manifold. Adiabatic decomposition K7 ≈ K3 ×ばつ T2 ×ばつ I reduces 7D PDEs to 1D Sturm-Liouville ODEs. All Betti numbers confirmed spectrally: b0=1, b1=0, b2=21, b3=77. SD/ASD gap in K3 intersection matrix: ×ばつ.

• 1744 distinct eigenvalues (λ < 20)
• 4460 states with multiplicities
• Three-scale hierarchy: neck, T2, K3

1. Introduction — Context, adiabatic ansatz validation
2. The Metric — Chebyshev-Cholesky summary, certification
3. Scalar Laplacian — Spectral gap, Weyl law, KK tower
4. Hodge Laplacian on 2-Forms — b2=21 confirmation, SD/ASD structure
5. Harmonic Forms & Betti Numbers — K3 forms, K7 assembly, b3=77
6. 1-Form Hodge Laplacian — Spectral democracy to 10−4, b1=0
7. Singular Limits — ADE singularity model, spectral stability
8. Discussion — G2-MSSM, F-theory, string landscape
9. Conclusion

1. Metric profiles: neck transition and ACyl decay
2. Scalar eigenvalue staircase (Weyl law)
3. First 5 scalar eigenfunctions
4. T2 channel spectra (adiabatic additivity)
5. 2-form spectrum with ×ばつ gap

• Paper Explicit G2 Metric — The metric this paper analyzes
• Paper Main Framework — Physics predictions from this geometry
• Paper S1 Foundations — TCS construction theory
• Observable Reference — Predictions catalog