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Paper S1 Foundations
Brieuc de La Fournière edited this page Mar 12, 2026
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Supplement S1: Mathematical Foundations — E8 Exceptional Lie Algebra, G2 Holonomy Manifolds, and K7 Construction
Brieuc de La Fournière (2026) Full text (markdown) | Zenodo DOI: 10.5281/zenodo.18837071
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.
| 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 ×ばつかける 35 ×ばつかける 52 ×ばつかける 7 =わ 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 |
- 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
Weyl = (dim(G2)+1)/N_gen = b2/N_gen − p2 = dim(G2) − rank(E8) − 1 = 5
- Paper Main Framework — Main paper
- Paper S2 Derivations — All 33 derivations
- Paper Explicit G2 Metric — Numerical metric
- Glossary — Term definitions
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