Comprehensive definitions of technical terms, mathematical notation, and acronyms used in the GIFT framework.
Geometric Information Field Theory. The framework proposing that fundamental physics parameters emerge as topological invariants from E₈×E₈ structure compactified on G₂ holonomy manifolds.
A compact 7-dimensional Riemannian manifold with G₂ holonomy. The subscript 7 indicates dimension. Specific topological properties: b₂(K₇) = 21, b₃(K₇) = 77.
Framework uses hierarchical classification for results:
The largest exceptional simple Lie algebra. Properties:
Product of two independent copies of E₈. Total dimension 496 = 2×248.
The automorphism group of the octonions. A 14-dimensional exceptional Lie group. Smallest exceptional group, important for 7-dimensional geometry.
Special Unitary Group. Group of N×N unitary matrices with determinant 1.
Five Lie algebras that don’t fit standard infinite families:
Symmetry group of root system. For E₈, the Weyl group has order 696,729,600.
Set of vectors in Euclidean space satisfying reflection symmetry. For E₈: 240 roots arranged in highly symmetric configuration.
Dimension of maximal torus (maximal abelian subgroup). For E₈: rank 8.
Geometric property describing how vectors change when parallel-transported around closed loops. G₂ holonomy implies special geometric structure.
Mathematical tool measuring topological features. For K₇:
Topological invariants counting independent homology classes.
Anti-de Sitter space in 4 dimensions. Maximally symmetric spacetime with negative cosmological constant. Used in holographic models and GIFT compactification.
Topological space that is closed (contains all limit points) and bounded. K₇ is compact, allowing consistent dimensional reduction.
Manifold with Ricci curvature tensor equal to zero. G₂ holonomy manifolds are automatically Ricci-flat, suitable for compactification.
Differential forms satisfying Laplace equation. Zero modes in dimensional reduction corresponding to 4D fields.
Process of compactifying extra dimensions to derive lower-dimensional effective theory. Fields decompose into tower of modes.
Current theory of particle physics describing electromagnetic, weak, and strong forces. Contains 19 free parameters in conventional formulation.
Family of fermions. Standard Model has three generations:
Strength of force interaction. Three in Standard Model:
Dimensionless coupling for electromagnetic interaction. α ≈ 1/137.036.
Parameter relating electromagnetic and weak forces. sin²θ_W ≈ 0.231.
Coupling constant for quantum chromodynamics. α_s(M_Z) ≈ 0.118.
Cabibbo-Kobayashi-Maskawa matrix. 3×3 unitary matrix describing quark mixing between generations. Contains 4 independent parameters (3 angles, 1 phase).
Pontecorvo-Maki-Nakagawa-Sakata matrix. Analogous to CKM for neutrino sector. Contains 3 mixing angles (θ₁₂, θ₁₃, θ₂₃) and CP violation phase (δ_CP).
Breaking of combined charge conjugation (C) and parity (P) symmetry. Observed in quark and neutrino sectors.
Interaction strength between fermions and Higgs field, determining fermion masses.
Process by which gauge bosons acquire mass through spontaneous symmetry breaking.
Vacuum Expectation Value. Non-zero value of Higgs field in vacuum, v ≈ 246 GeV.
Number of fermion generations. Experimentally: 3. GIFT predicts: rank(E₈) - rank(Weyl) = 3.
CP-violating phase in neutrino mixing. GIFT predicts: 197° from formula 7·dim(G₂) + ζ(3) + √5.
Three neutrino mixing angles in PMNS matrix.
Parameter in Koide formula relating charged lepton masses:
Q = (mₑ + mμ + mτ)² / (mₑ² + mμ² + mτ²)
Experimental: Q ≈ 2/3. GIFT: Q = 2/3 exactly.
Dark energy density as fraction of critical density. Experimental: Ω_DE ≈ 0.689. GIFT: Ω_DE = ln(2) ≈ 0.693.
Hubble constant. Current expansion rate of universe. Local measurements: ~73 km/s/Mpc. CMB measurements: ~67 km/s/Mpc. “Hubble tension” refers to discrepancy.
Pi, ratio of circle circumference to diameter. π ≈ 3.14159…
Euler’s number, base of natural logarithm. e ≈ 2.71828…
Golden ratio. φ = (1 + √5)/2 ≈ 1.61803… Appears in E₈ via McKay correspondence.
Riemann zeta function at 3. ζ(3) = 1 + 1/8 + 1/27 + … ≈ 1.202. Appears in δ_CP formula.
Natural logarithm of 2 ≈ 0.693. Related to dark energy density and information theory (bits vs nats).
Structure based on powers of 2. E₈×E₈ dimension 496 ≈ 2⁴⁸ + 48, suggestive of binary encoding.
Method of protecting quantum information from errors. E₈ lattice forms [[248, 12, 56]] code.
Notation for quantum error correction code:
Ratio of input to output dimensions. E₈×E₈: 496 → 99 gives ratio ≈ 5:1.
Particle Data Group. International collaboration compiling experimental particle physics data. Standard reference for measured parameters.
Large Hadron Collider. Particle accelerator at CERN. Discovered Higgs boson in 2012.
Particle physics experiment in Japan studying B mesons and precision measurements.
Deep Underground Neutrino Experiment. Future neutrino experiment in USA for precision measurements.
Tokai to Kamioka. Long-baseline neutrino experiment in Japan.
NuMI Off-Axis νₑ Appearance. Neutrino experiment in USA.
Dimension of Lie group or algebra G. Example: dim(E₈) = 248.
Rank of Lie algebra G. Example: rank(E₈) = 8.
n-th cohomology group of manifold M. Example: H²(K₇) = ℝ²¹.
n-th Betti number, dimension of Hⁿ. Example: b₃(K₇) = 77.
Absolute value or cardinality (size of set).
Direct sum of vector spaces or algebras.
Tensor product.
“Element of” (set membership).
“For all” (universal quantifier).
“There exists” (existential quantifier).
Approximately equal.
Identically equal or defined as.
Of the same order of magnitude, or equivalent to.
Fine structure constant, α ≈ 1/137.
Base coupling parameter, β₀ = 1/(4π²).
Heat kernel coefficient or Euler-Mascheroni constant.
CP violation phase (δ_CP) or small deviation.
Symmetry breaking parameter, ε₀ = 1/8.
Riemann zeta function, ζ(3) appears in δ_CP.
Mixing angles (θ₁₂, θ₁₃, θ₂₃, θ_W).
Correlation parameter, ξ = 5β₀/2.
Density parameters (Ω_DE for dark energy).
Golden ratio or angle.
Geometric Information Field Theory
Standard Model (of particle physics)
Grand Unified Theory
Quantum Chromodynamics (strong force)
Quantum Electrodynamics (electromagnetic force)
Electroweak (unified electromagnetic and weak theory)
Supersymmetry
Cosmic Microwave Background
Vacuum Expectation Value
Renormalization Group
Ultraviolet (high energy)
Infrared (low energy)
Leading Order / Next-to-Leading Order / Next-to-Next-to-Leading Order (perturbative expansion)
System where ℏ = c = 1. Energies, masses, and momenta have same dimensions.
Giga-electron-volt, 10⁹ eV. Common energy/mass unit in particle physics.
Mega-electron-volt, 10⁶ eV.
Mathematical claim with rigorous proof provided.
Direct consequence of manifold topology.
Zero deviation by mathematical construction.
Experimental agreement <1% deviation.
Calculation or result pending refinement.
For more detailed explanations:
docs/FAQ.mdThis glossary is continuously updated. To suggest additions or corrections:
Last updated: v2.0.0 (2025-10-24)