Paper S1 Foundations

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) | Zenodo DOI: 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.

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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

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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

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The Weyl Triple Identity

Weyl = (dim(G₂)+1)/N_gen = b₂/N_gen − p₂ = dim(G₂) − rank(E₈) − 1 = 5

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Related

• Paper-Main-Framework — Main paper
• Paper-S2-Derivations — All 33 derivations
• Paper-Explicit-G2-Metric — Numerical metric
• Glossary — Term definitions