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 for 7D Cross Product
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
Direct Primes
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
TCS Spectral Bounds
▶
14.1
TCS Manifold Structure
14.2
Spectral Bound Constants
14.3
Model Theorem
14.4
Literature Axioms (Langlais/CGN)
14.5
Connection to Mass Gap
15
Physical Spectral Gap
▶
15.1
Parallel Spinors (Berger Classification)
15.2
Axiom-Free Derivation of 13/99
16
Selberg Trace Formula Bridge
▶
16.1
Trace Formula (Standard Result)
16.2
Geodesic-Prime Correspondence
16.3
Cross-Module Identities (Proven)
17
Mollified Dirichlet Polynomial
▶
17.1
Cosine-Squared Mollifier Kernel
17.2
Mollified Dirichlet Polynomial
17.3
Adaptive Cutoff
17.4
Main Results (Numerical Certificates)
18
GIFT-Zeta Correspondences
▶
18.1
Axiomatized Zeta Zero Sequence
18.2
Primary Correspondences
18.3
Relative Precision
18.4
Spectral Interpretation
18.5
Multiples of \(\mathrm{dim}(K_7) = 7\)
18.6
Master Certificate
19
Summary and Status
▶
19.1
Proof Status Overview
19.2
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 for 7D Cross Product
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
Direct Primes
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
TCS Spectral Bounds
14.1
TCS Manifold Structure
14.2
Spectral Bound Constants
14.3
Model Theorem
14.4
Literature Axioms (Langlais/CGN)
14.5
Connection to Mass Gap
15
Physical Spectral Gap
15.1
Parallel Spinors (Berger Classification)
15.2
Axiom-Free Derivation of 13/99
16
Selberg Trace Formula Bridge
16.1
Trace Formula (Standard Result)
16.2
Geodesic-Prime Correspondence
16.3
Cross-Module Identities (Proven)
17
Mollified Dirichlet Polynomial
17.1
Cosine-Squared Mollifier Kernel
17.2
Mollified Dirichlet Polynomial
17.3
Adaptive Cutoff
17.4
Main Results (Numerical Certificates)
18
GIFT-Zeta Correspondences
18.1
Axiomatized Zeta Zero Sequence
18.2
Primary Correspondences
18.3
Relative Precision
18.4
Spectral Interpretation
18.5
Multiples of \(\mathrm{dim}(K_7) = 7\)
18.6
Master Certificate
19
Summary and Status
19.1
Proof Status Overview
19.2
Key Results