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
Joyce Existence Theorem
▶
7.1
PINN Verification
7.2
Existence
8
Analytical Metric Extraction
▶
8.1
GIFT-Native PINN Architecture
8.2
Certified Bounds
8.3
Analytical Extraction
9
Explicit G2 Metric
▶
9.1
The Standard G2 3-form
9.2
Linear Index Representation
9.3
The GIFT Scale Factor
9.4
The Explicit Metric
9.5
Torsion Vanishes
9.6
Summary
10
169-Parameter Chebyshev-Cholesky Metric
▶
10.1
Chebyshev-Cholesky Parametrization
10.2
SPD Structure and Determinant
10.3
Newton-Kantorovich Certification
10.4
K3 Harmonic Correction and Torsion Classes
10.5
K3 Newton-Kantorovich Certificate (CI(2,2,2))
11
TCS Spectral Bounds
▶
11.1
TCS Manifold Structure
11.2
Spectral Bound Constants
11.3
Model Theorem
11.4
Literature Axioms (Langlais/CGN)
11.5
Connection to Mass Gap
12
Physical Spectral Gap
▶
12.1
Parallel Spinors (Berger Classification)
12.2
Axiom-Free Derivation of 13/99
13
G\(_2\) Metric Properties
▶
13.1
Non-Flatness of \(K_7\) (Bieberbach Bound)
13.2
Spectral Degeneracy Pattern \([1, 10, 9, 30]\)
13.3
SPD\(_7\) Metric Parametrization
13.4
Triple Derivation of \(\det (g) = 65/32\)
13.5
\(\kappa _T^{-1} = 61\) Structural Decomposition
13.6
Master Certificate
14
TCS Piecewise Metric Structure
▶
14.1
Building Block Asymmetry
14.2
Effective Degrees of Freedom per Block
14.3
Matrix Space Decomposition
14.4
Fano Plane Automorphism Group
14.5
Fano Incidence Arithmetic
14.6
Kovalev Involution Eigenspace
14.7
Master Certificate
15
Conformal Rigidity
▶
15.1
\(G_2\) Representation Decompositions
15.2
Conformal Rigidity Theorem
15.3
Moduli Space Structure
15.4
Master Certificate
16
Spectral Scaling on the TCS Neck
▶
16.1
Neumann Eigenvalue Hierarchy
16.2
Neck Length Ratio and Selection Constant
16.3
Division Algorithm Connection
16.4
Sub-Gap Mode Counting
16.5
Second Eigenvalue and \(E_7\)
16.6
Pell Equation
16.7
Spectral-Topological Dictionary
16.8
Master Certificate
17
Poincaré Duality and the GIFT Spectrum
▶
17.1
Full Betti Sequence and Total Betti Number
17.2
Structural Identity: \(H^*= 1 + 2 \mathrm{dim}(K_7)^2\)
17.3
Holonomy Embedding Chain \(G_2{\lt} \mathrm{SO}(7) {\lt} \mathrm{GL}(7)\)
17.4
\(G_2\) Torsion Decomposition
17.5
SU(3) Branching Rule
17.6
Betti–Torsion Bridge
17.7
Master Certificate
18
\(G_2\) Differential Geometry
▶
18.1
Differential Forms on \(\mathbb {R}^7\)
18.2
Hodge Star on \(\mathbb {R}^7\)
18.3
Master Certificate
19
Dimensional Hierarchy
▶
19.1
Cohomological Suppression
19.2
Vacuum Structure
19.3
\(E_8\to E_6\) Symmetry Cascade
19.4
Absolute Lepton Masses
20
Extended Observables
▶
20.1
Weak Mixing Angle (14 Expressions)
20.2
PMNS Neutrino Mixing
20.3
Quark Mass Ratios
20.4
Boson Mass Ratios
20.5
CKM Quark Mixing
20.6
Cosmological Parameters
20.7
Universality
21
Mass Gap Ratio and Universal Law
▶
21.1
Spectral Theory Foundations
21.2
Bare Algebraic Ratio: 14/99
21.3
Universal Spectral Law
22
Selection Principle and Yang–Mills
▶
22.1
Selection Constant
22.2
Cheeger–Buser Inequality
22.3
KK Spectral Bridge and Yang–Mills
23
Summary and Status
▶
23.1
Proof Status Overview
23.2
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
Joyce Existence Theorem
7.1
PINN Verification
7.2
Existence
8
Analytical Metric Extraction
8.1
GIFT-Native PINN Architecture
8.2
Certified Bounds
8.3
Analytical Extraction
9
Explicit G2 Metric
9.1
The Standard G2 3-form
9.2
Linear Index Representation
9.3
The GIFT Scale Factor
9.4
The Explicit Metric
9.5
Torsion Vanishes
9.6
Summary
10
169-Parameter Chebyshev-Cholesky Metric
10.1
Chebyshev-Cholesky Parametrization
10.2
SPD Structure and Determinant
10.3
Newton-Kantorovich Certification
10.4
K3 Harmonic Correction and Torsion Classes
10.5
K3 Newton-Kantorovich Certificate (CI(2,2,2))
11
TCS Spectral Bounds
11.1
TCS Manifold Structure
11.2
Spectral Bound Constants
11.3
Model Theorem
11.4
Literature Axioms (Langlais/CGN)
11.5
Connection to Mass Gap
12
Physical Spectral Gap
12.1
Parallel Spinors (Berger Classification)
12.2
Axiom-Free Derivation of 13/99
13
G\(_2\) Metric Properties
13.1
Non-Flatness of \(K_7\) (Bieberbach Bound)
13.2
Spectral Degeneracy Pattern \([1, 10, 9, 30]\)
13.3
SPD\(_7\) Metric Parametrization
13.4
Triple Derivation of \(\det (g) = 65/32\)
13.5
\(\kappa _T^{-1} = 61\) Structural Decomposition
13.6
Master Certificate
14
TCS Piecewise Metric Structure
14.1
Building Block Asymmetry
14.2
Effective Degrees of Freedom per Block
14.3
Matrix Space Decomposition
14.4
Fano Plane Automorphism Group
14.5
Fano Incidence Arithmetic
14.6
Kovalev Involution Eigenspace
14.7
Master Certificate
15
Conformal Rigidity
15.1
\(G_2\) Representation Decompositions
15.2
Conformal Rigidity Theorem
15.3
Moduli Space Structure
15.4
Master Certificate
16
Spectral Scaling on the TCS Neck
16.1
Neumann Eigenvalue Hierarchy
16.2
Neck Length Ratio and Selection Constant
16.3
Division Algorithm Connection
16.4
Sub-Gap Mode Counting
16.5
Second Eigenvalue and \(E_7\)
16.6
Pell Equation
16.7
Spectral-Topological Dictionary
16.8
Master Certificate
17
Poincaré Duality and the GIFT Spectrum
17.1
Full Betti Sequence and Total Betti Number
17.2
Structural Identity: \(H^*= 1 + 2 \mathrm{dim}(K_7)^2\)
17.3
Holonomy Embedding Chain \(G_2{\lt} \mathrm{SO}(7) {\lt} \mathrm{GL}(7)\)
17.4
\(G_2\) Torsion Decomposition
17.5
SU(3) Branching Rule
17.6
Betti–Torsion Bridge
17.7
Master Certificate
18
\(G_2\) Differential Geometry
18.1
Differential Forms on \(\mathbb {R}^7\)
18.2
Hodge Star on \(\mathbb {R}^7\)
18.3
Master Certificate
19
Dimensional Hierarchy
19.1
Cohomological Suppression
19.2
Vacuum Structure
19.3
\(E_8\to E_6\) Symmetry Cascade
19.4
Absolute Lepton Masses
20
Extended Observables
20.1
Weak Mixing Angle (14 Expressions)
20.2
PMNS Neutrino Mixing
20.3
Quark Mass Ratios
20.4
Boson Mass Ratios
20.5
CKM Quark Mixing
20.6
Cosmological Parameters
20.7
Universality
21
Mass Gap Ratio and Universal Law
21.1
Spectral Theory Foundations
21.2
Bare Algebraic Ratio: 14/99
21.3
Universal Spectral Law
22
Selection Principle and Yang–Mills
22.1
Selection Constant
22.2
Cheeger–Buser Inequality
22.3
KK Spectral Bridge and Yang–Mills
23
Summary and Status
23.1
Proof Status Overview
23.2
Key Results