Formally verified mathematical relations from the GIFT framework. All theorems proven in Lean 4 and Coq.
The GIFT Certificate proves 180+ mathematical identities organized in five foundational pillars:
dim(E₈) = 248 = 240 roots + 8 rank
= 8 × 31 (Mersenne structure)
= 120 + 128 (SO(16) decomposition)
- Complete root enumeration: 112 (D₈) + 128 (half-integer)
- Weyl group order: 2¹⁴ × 3⁵ × 5² × 7 = 696,729,600
- Weyl reflection preserves E₈ lattice
dim(G₂) = 14 = 12 roots + 2 rank
= GL(7) orbit stabilizer: 49 - 35
- 7D cross product with Lagrange identity: ‖u × v‖² = ‖u‖²‖v‖² - ⟨u,v⟩²
- Fano plane structure (7 lines ↔ 7 octonion imaginaries)
- Bilinearity, antisymmetry, octonion structure proven
M₁ = Quintic in CP⁴: b₂ = 11, b₃ = 40
M₂ = CI(2,2,2) in CP⁶: b₂ = 10, b₃ = 37
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K₇ = M₁ #_TCS M₂: b₂ = 21, b₃ = 77 (BOTH DERIVED!)
H* = b₂ + b₃ + 1 = 99
- TCS (Twisted Connected Sum) construction from Corti-Haskins-Nordström-Pacini
- Both Betti numbers now derived from building blocks (was: b₃ input)
- Hodge duality and Poincaré duality verified
K₇ admits torsion-free G₂ structure
‖T‖ < 0.00141 vs threshold 0.0288 (20× margin)
- Banach fixed-point formalization
- Sobolev embedding H⁴ -> C⁰ (4 > 7/2)
- Implicit function theorem conditions verified
Weyl Triple Identity: 3 independent paths to Weyl = 5
(dim_G₂ + 1) / N_gen = 5
b₂ / N_gen - p₂ = 5
dim_G₂ - rank_E₈ - 1 = 5
PSL(2,7) = 168: Fano plane symmetry
(b₃ + dim_G₂) + b₃ = 168
rank_E₈ × b₂ = 168
N_gen × (b₃ - b₂) = 168
The Certificate derives Standard Model parameters from topology:
| Relation | Formula | Value |
|---|---|---|
| Weinberg angle | sin²θ_W = 3(b₃+dim_G₂)/(13×b₂) | 3/13 |
| Koide parameter | Q = 2×dim_G₂/(3×b₂) | 2/3 |
| Generation count | N_gen | 3 |
| κ_T denominator | b₃ - dim_G₂ - p₂ | 61 |
| γ_GIFT | (2×rank_E₈ + 5×H*)/(10×dim_G₂ + 3×dim_E₈) | 511/884 |
| Ω_DE | (b₂ + b₃)/H* | 98/99 |
| m_τ/m_e | (b₃ - b₂) × 62 + 5 | 3477 |
See Lean/GIFT/Certificate.lean for complete theorem statements.
- Sequence Embeddings: Fibonacci F₃–F₁₂ and Lucas L₀–L₉ map to GIFT constants
- Prime Atlas: 100% coverage of primes < 200 via three generators (b₃, H*, dim_E₈)
- Monstrous Moonshine: 196883 = 47 × 59 × 71, j-invariant 744 = 3 × dim_E₈
- McKay Correspondence: E₈ ↔ Binary Icosahedral ↔ Golden Ratio
pip install giftpyfrom gift_core import *
# Certified constants
print(SIN2_THETA_W) # Fraction(3, 13)
print(KAPPA_T) # Fraction(1, 61)
print(GAMMA_GIFT) # Fraction(511, 884)# Lean 4
cd Lean && lake build
# Coq
cd COQ && makeBlueprint structure inspired by KakeyaFiniteFields.
MIT
GIFT Core v3.2.0
