giftpy

GIFT Core

Formally verified mathematical relations from the GIFT framework. All theorems proven in Lean 4 and Coq.

Certificate Structure

The GIFT Certificate proves 180+ mathematical identities organized in five foundational pillars:

1. E₈ Root System (248 dimensions)

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

2. G₂ Holonomy (14 dimensions)

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

3. K₇ Manifold via TCS (v3.2)

M₁ = Quintic in CP⁴:    b₂ = 11,  b₃ = 40
M₂ = CI(2,2,2) in CP⁶:  b₂ = 10,  b₃ = 37
─────────────────────────────────────────
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

4. Joyce Existence Theorem

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

5. Structural Identities (v3.2)

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

Physical Relations

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.

Extensions

• 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

Installation

pip install giftpy

Quick Start

from gift_core import *

# Certified constants
print(SIN2_THETA_W)   # Fraction(3, 13)
print(KAPPA_T)        # Fraction(1, 61)
print(GAMMA_GIFT)     # Fraction(511, 884)

Building Proofs

# Lean 4
cd Lean && lake build

# Coq
cd COQ && make

Documentation

• Changelog
• Usage Guide
• Full Framework

Acknowledgments

Blueprint structure inspired by KakeyaFiniteFields.

License

MIT

GIFT Core v3.2.0