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
169-Parameter Chebyshev-Cholesky Metric (Phase 1)
▶
14.1
Chebyshev-Cholesky Parametrization
14.2
SPD Structure and Determinant
14.3
Newton-Kantorovich Certification
14.4
K3 Harmonic Correction and Torsion Classes
15
TCS Spectral Bounds
▶
15.1
TCS Manifold Structure
15.2
Spectral Bound Constants
15.3
Model Theorem
15.4
Literature Axioms (Langlais/CGN)
15.5
Connection to Mass Gap
16
Physical Spectral Gap
▶
16.1
Parallel Spinors (Berger Classification)
16.2
Axiom-Free Derivation of 13/99
17
Selberg Trace Formula Bridge
▶
17.1
Trace Formula (Standard Result)
17.2
Geodesic-Prime Correspondence
17.3
Cross-Module Identities (Proven)
18
Mollified Dirichlet Polynomial
▶
18.1
Cosine-Squared Mollifier Kernel
18.2
Mollified Dirichlet Polynomial
18.3
Adaptive Cutoff
18.4
Main Results (Numerical Certificates)
19
GIFT-Zeta Correspondences
▶
19.1
Axiomatized Zeta Zero Sequence
19.2
Primary Correspondences
19.3
Relative Precision
19.4
Spectral Interpretation
19.5
Multiples of \(\mathrm{dim}(K_7) = 7\)
19.6
Master Certificate
20
G\(_2\) Metric Properties
▶
20.1
Non-Flatness of \(K_7\) (Bieberbach Bound)
20.2
Spectral Degeneracy Pattern \([1, 10, 9, 30]\)
20.3
SPD\(_7\) Metric Parametrization
20.4
Triple Derivation of \(\det (g) = 65/32\)
20.5
\(\kappa _T^{-1} = 61\) Structural Decomposition
20.6
Master Certificate
21
TCS Piecewise Metric Structure
▶
21.1
Building Block Asymmetry
21.2
Effective Degrees of Freedom per Block
21.3
Matrix Space Decomposition
21.4
Fano Plane Automorphism Group
21.5
Fano Incidence Arithmetic
21.6
Kovalev Involution Eigenspace
21.7
Master Certificate
22
Conformal Rigidity
▶
22.1
\(G_2\) Representation Decompositions
22.2
Conformal Rigidity Theorem
22.3
Moduli Space Structure
22.4
Master Certificate
23
Spectral Scaling on the TCS Neck
▶
23.1
Neumann Eigenvalue Hierarchy
23.2
Neck Length Ratio and Selection Constant
23.3
Division Algorithm Connection
23.4
Sub-Gap Mode Counting
23.5
Second Eigenvalue and \(E_7\)
23.6
Pell Equation
23.7
Spectral-Topological Dictionary
23.8
Master Certificate
24
Poincaré Duality and the GIFT Spectrum
▶
24.1
Full Betti Sequence and Total Betti Number
24.2
Structural Identity: \(H^*= 1 + 2 \mathrm{dim}(K_7)^2\)
24.3
Holonomy Embedding Chain \(G_2{\lt} \mathrm{SO}(7) {\lt} \mathrm{GL}(7)\)
24.4
\(G_2\) Torsion Decomposition
24.5
SU(3) Branching Rule
24.6
Betti–Torsion Bridge
24.7
Master Certificate
25
\(G_2\) Differential Geometry
▶
25.1
Differential Forms on \(\mathbb {R}^7\)
25.2
Hodge Star on \(\mathbb {R}^7\)
25.3
Master Certificate
26
Dimensional Hierarchy
▶
26.1
Cohomological Suppression
26.2
Vacuum Structure
26.3
\(E_8\to E_6\) Symmetry Cascade
26.4
Absolute Lepton Masses
27
Extended Observables
▶
27.1
Weak Mixing Angle (14 Expressions)
27.2
PMNS Neutrino Mixing
27.3
Quark Mass Ratios
27.4
Boson Mass Ratios
27.5
CKM Quark Mixing
27.6
Cosmological Parameters
27.7
Universality
28
Mass Gap Ratio and Universal Law
▶
28.1
Spectral Theory Foundations
28.2
Bare Mass Gap Ratio: 14/99
28.3
Universal Spectral Law
29
Selection Principle and Yang–Mills
▶
29.1
Selection Constant
29.2
Cheeger–Buser Inequality
29.3
Yang–Mills Mass Gap
29.4
Connes Bridge
30
Summary and Status
▶
30.1
Proof Status Overview
30.2
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
169-Parameter Chebyshev-Cholesky Metric (Phase 1)
14.1
Chebyshev-Cholesky Parametrization
14.2
SPD Structure and Determinant
14.3
Newton-Kantorovich Certification
14.4
K3 Harmonic Correction and Torsion Classes
15
TCS Spectral Bounds
15.1
TCS Manifold Structure
15.2
Spectral Bound Constants
15.3
Model Theorem
15.4
Literature Axioms (Langlais/CGN)
15.5
Connection to Mass Gap
16
Physical Spectral Gap
16.1
Parallel Spinors (Berger Classification)
16.2
Axiom-Free Derivation of 13/99
17
Selberg Trace Formula Bridge
17.1
Trace Formula (Standard Result)
17.2
Geodesic-Prime Correspondence
17.3
Cross-Module Identities (Proven)
18
Mollified Dirichlet Polynomial
18.1
Cosine-Squared Mollifier Kernel
18.2
Mollified Dirichlet Polynomial
18.3
Adaptive Cutoff
18.4
Main Results (Numerical Certificates)
19
GIFT-Zeta Correspondences
19.1
Axiomatized Zeta Zero Sequence
19.2
Primary Correspondences
19.3
Relative Precision
19.4
Spectral Interpretation
19.5
Multiples of \(\mathrm{dim}(K_7) = 7\)
19.6
Master Certificate
20
G\(_2\) Metric Properties
20.1
Non-Flatness of \(K_7\) (Bieberbach Bound)
20.2
Spectral Degeneracy Pattern \([1, 10, 9, 30]\)
20.3
SPD\(_7\) Metric Parametrization
20.4
Triple Derivation of \(\det (g) = 65/32\)
20.5
\(\kappa _T^{-1} = 61\) Structural Decomposition
20.6
Master Certificate
21
TCS Piecewise Metric Structure
21.1
Building Block Asymmetry
21.2
Effective Degrees of Freedom per Block
21.3
Matrix Space Decomposition
21.4
Fano Plane Automorphism Group
21.5
Fano Incidence Arithmetic
21.6
Kovalev Involution Eigenspace
21.7
Master Certificate
22
Conformal Rigidity
22.1
\(G_2\) Representation Decompositions
22.2
Conformal Rigidity Theorem
22.3
Moduli Space Structure
22.4
Master Certificate
23
Spectral Scaling on the TCS Neck
23.1
Neumann Eigenvalue Hierarchy
23.2
Neck Length Ratio and Selection Constant
23.3
Division Algorithm Connection
23.4
Sub-Gap Mode Counting
23.5
Second Eigenvalue and \(E_7\)
23.6
Pell Equation
23.7
Spectral-Topological Dictionary
23.8
Master Certificate
24
Poincaré Duality and the GIFT Spectrum
24.1
Full Betti Sequence and Total Betti Number
24.2
Structural Identity: \(H^*= 1 + 2 \mathrm{dim}(K_7)^2\)
24.3
Holonomy Embedding Chain \(G_2{\lt} \mathrm{SO}(7) {\lt} \mathrm{GL}(7)\)
24.4
\(G_2\) Torsion Decomposition
24.5
SU(3) Branching Rule
24.6
Betti–Torsion Bridge
24.7
Master Certificate
25
\(G_2\) Differential Geometry
25.1
Differential Forms on \(\mathbb {R}^7\)
25.2
Hodge Star on \(\mathbb {R}^7\)
25.3
Master Certificate
26
Dimensional Hierarchy
26.1
Cohomological Suppression
26.2
Vacuum Structure
26.3
\(E_8\to E_6\) Symmetry Cascade
26.4
Absolute Lepton Masses
27
Extended Observables
27.1
Weak Mixing Angle (14 Expressions)
27.2
PMNS Neutrino Mixing
27.3
Quark Mass Ratios
27.4
Boson Mass Ratios
27.5
CKM Quark Mixing
27.6
Cosmological Parameters
27.7
Universality
28
Mass Gap Ratio and Universal Law
28.1
Spectral Theory Foundations
28.2
Bare Mass Gap Ratio: 14/99
28.3
Universal Spectral Law
29
Selection Principle and Yang–Mills
29.1
Selection Constant
29.2
Cheeger–Buser Inequality
29.3
Yang–Mills Mass Gap
29.4
Connes Bridge
30
Summary and Status
30.1
Proof Status Overview
30.2
Key Results