Paper: S1: Mathematical Foundations
Supplement S1: Mathematical Foundations: E₈ Exceptional Lie Algebra, G₂ Holonomy Manifolds, and K₇ Construction
Brieuc de La Fournière (2026) Full text (markdown, v3.4) | v3.3 archive on Zenodo: 10.5281/zenodo.18837071
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Abstract
Develops E₈ architecture, G₂ holonomy manifolds via kernel of Lie derivative, and K₇ construction via twisted connected sum. Establishes algebraic reference form det(g) = 65/32 and Joyce existence theorem guaranteeing torsion-free metric.
Key Results
| Result | Value | Status |
|---|---|---|
| Division algebra chain | ℝ(1) → ℂ(2) → ℍ(4) → 𝕆(8) | Terminal at 8 |
| E₈ root system | 240 roots = 112 D₈ + 128 half-integer | Verified |
| |W(E₈)| | 2¹⁴ × 3⁵ × 5² × 7 = 696,729,600 | Lean-verified |
| TCS building blocks | M₁(quintic)[b₂=11,b₃=40] + M₂(CI(2,2,2))[b₂=10,b₃=37] | → K₇[21,77] |
| det(g) | 65/32 (3 independent paths) | Exact |
| Spectral gap | λ₁ = 13/99 | Algebraic |
Section Structure
- Part 0: Octonionic Foundation: Why 𝕆 is terminal, G₂ = Aut(𝕆), Fano plane
- Part I: E₈ Exceptional Lie Algebra: Root system, Weyl group, exceptional chain
- Part II: G₂ Holonomy Manifolds: Definition, Berger classification, torsion classes W₁–W₂₇
- Part III: K₇ Manifold Construction: TCS framework, ACyl building blocks, Mayer-Vietoris
- Part IV: Metric Structure & Verification, κ_T = 1/61, det(g) = 65/32, Joyce existence
The Weyl Triple Identity
Weyl = (dim(G₂)+1)/N_gen = b₂/N_gen − p₂ = dim(G₂) − rank(E₈) − 1 = 5
Related
- Paper Main Framework: Main paper
- Paper S2 Derivations: All 33 derivations
- Paper Explicit G2 Metric: Numerical metric
- Glossary: Term definitions