Paper: Explicit G₂ Metric

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.19892350

Abstract

Constructs explicit 169-parameter Chebyshev-Cholesky metric on compact TCS K₇. Newton-Kantorovich certificate proves unique torsion-free G₂ metric g* exists within distance 4.86×10⁻⁶. Initial torsion ‖T‖ = 8.94×10⁻² reduced to 2.98×10⁻⁵ in 5 Joyce iterations (3000× reduction).

Key Results

Certification Chain

Quantity Value
Initial torsion ‖T‖₀ 8.936×10⁻²
Final torsion ‖T‖₅ 2.984×10⁻⁵
Reduction factor 2995×
NK contraction h 6.65×10⁻⁸
NK threshold 0.5
Safety margin ×7.5M
Distance to exact metric ≤ 4.86×10⁻⁶

Metric Properties

Property Value
Parameters 169 (168 Chebyshev + 1 ACyl decay)
det(g) 65/32 (exact)
|φ|² 42 (error < 10⁻¹⁴)
Holonomy Hol(g*) = G₂
Torsion class 99.6% in W₃, |dφ|²/|d*φ|² = 1/5

Eigenvalue Hierarchy

Three-scale structure:

Section Structure

  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)

Figures