An Explicit Approximate G2 Metric on a Compact TCS 7-Manifold with Certified Torsion-Free Completion

Brieuc de La Fournière (2026) Full text (markdown) | Zenodo DOI: 10.5281/zenodo.18860358

Constructs explicit 169-parameter Chebyshev-Cholesky metric on compact TCS K7. Newton-Kantorovich certificate proves unique torsion-free G2 metric g* exists within distance ×ばつ10−6. Initial torsion ‖T‖ = ×ばつ10−2 reduced to ×ばつ10−5 in 5 Joyce iterations (×ばつ reduction).

Three-scale structure:

• Neck (seam): λ0 ≈ 6.8
• T2 (fiber): λ1,6 ≈ 2.9
• K3 (fiber): λ2−5 ≈ 1.1

1. Introduction — Context, objective, scope & claims
2. The Manifold — TCS construction, topology (b2=21, b3=77)
3. The Metric — Model hierarchy, coordinates, Chebyshev-Cholesky parametrization
4. Norm Definitions & Domain — Metric distance, torsion norms, NK norm
5. Torsion Analysis — Initial approximation, K3 verification, Gauss-Newton reduction
6. Certification — NK convergence, interval arithmetic, holonomy proof
7. Geometric Invariants — det(g)=65/32, |φ|2=42, Hol(g*)=G2
8. Discussion — Limitations, comparison with prior work
9. Reproducibility — Data files, companion notebook (< 1 min runtime)

• TCS visualization with torsion intensity coloring
• Atlas chart schematic
• Eigenvalue profile (three-scale hierarchy)
• Torsion convergence (log scale, 5 iterations)

• Paper Main Framework — Physics application
• Paper S1 Foundations — TCS construction theory
• Paper Spectral Geometry — Spectral analysis of this metric
• For Geometers — Computational pipeline overview