1
Introduction
2
Foundations: E8 Lattice
▶
2.1
Euclidean Space Setup
2.2
E8 Lattice Definition
2.3
Lattice Properties
3
Foundations: G2 Cross Product
▶
3.1
The Fano Plane
3.2
Cross Product Definition
3.3
Cross Product Properties
3.4
Lagrange Identity (B4)
4
Algebraic Foundations
▶
4.1
Octonion Structure
4.2
Betti Numbers from Octonions
5
SO(16) Decomposition
▶
5.1
SO(n) Dimension
5.2
Spinor Representation
5.3
Geometric and Spinorial Parts
5.4
Master Decomposition
6
Physical Relations
▶
6.1
Weinberg Angle
6.2
Koide Formula
6.3
Fine Structure Constant
6.4
Strong Coupling
6.5
Lepton Mass Ratios
6.6
Higgs Quartic
6.7
Cosmological Parameters
7
Fibonacci and Lucas Embeddings
▶
7.1
Fibonacci Embedding
7.2
Lucas Embedding
8
Prime Atlas
▶
8.1
Tier 1: Direct Constants
8.2
Heegner Numbers
9
Monstrous Moonshine
▶
9.1
Monster Dimension
9.2
j-Invariant
10
McKay Correspondence
▶
10.1
Icosahedral Structure
11
Joyce Existence Theorem
▶
11.1
PINN Verification
11.2
Existence
12
Analytical Metric Extraction
▶
12.1
GIFT-Native PINN Architecture
12.2
Certified Bounds
12.3
Analytical Extraction
13
Explicit G2 Metric
▶
13.1
The Standard G2 3-form
13.2
Linear Index Representation
13.3
The GIFT Scale Factor
13.4
The Explicit Metric
13.5
Torsion Vanishes
13.6
Summary
14
Summary and Status
▶
14.1
Proof Status Overview
14.2
Key Results
Dependency graph
GIFT: Geometric Information Field Theory
Brieuc de La Fournière
1
Introduction
2
Foundations: E8 Lattice
2.1
Euclidean Space Setup
2.2
E8 Lattice Definition
2.3
Lattice Properties
3
Foundations: G2 Cross Product
3.1
The Fano Plane
3.2
Cross Product Definition
3.3
Cross Product Properties
3.4
Lagrange Identity (B4)
4
Algebraic Foundations
4.1
Octonion Structure
4.2
Betti Numbers from Octonions
5
SO(16) Decomposition
5.1
SO(n) Dimension
5.2
Spinor Representation
5.3
Geometric and Spinorial Parts
5.4
Master Decomposition
6
Physical Relations
6.1
Weinberg Angle
6.2
Koide Formula
6.3
Fine Structure Constant
6.4
Strong Coupling
6.5
Lepton Mass Ratios
6.6
Higgs Quartic
6.7
Cosmological Parameters
7
Fibonacci and Lucas Embeddings
7.1
Fibonacci Embedding
7.2
Lucas Embedding
8
Prime Atlas
8.1
Tier 1: Direct Constants
8.2
Heegner Numbers
9
Monstrous Moonshine
9.1
Monster Dimension
9.2
j-Invariant
10
McKay Correspondence
10.1
Icosahedral Structure
11
Joyce Existence Theorem
11.1
PINN Verification
11.2
Existence
12
Analytical Metric Extraction
12.1
GIFT-Native PINN Architecture
12.2
Certified Bounds
12.3
Analytical Extraction
13
Explicit G2 Metric
13.1
The Standard G2 3-form
13.2
Linear Index Representation
13.3
The GIFT Scale Factor
13.4
The Explicit Metric
13.5
Torsion Vanishes
13.6
Summary
14
Summary and Status
14.1
Proof Status Overview
14.2
Key Results