giftpy 1.4.0

GIFT mathematical core - Formally verified constants (Lean 4 + Coq)

GIFT Core

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What is GIFT?

GIFT (Geometric Information Field Theory) is a speculative theoretical physics framework that attempts to derive Standard Model parameters from topological invariants of E8 x E8 gauge theory compactified on seven-dimensional manifolds with G2 holonomy.

This repository contains only the mathematical core: exact rational and integer relations that have been formally verified in two independent proof assistants. The physical interpretation of these relations remains conjectural and subject to experimental validation.

Important distinction: This package proves that certain mathematical identities hold exactly. Whether these identities correspond to physical reality is a separate empirical question.

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Installation

pip install giftpy

Usage

from gift_core import *

# Original 13 certified relations
print(SIN2_THETA_W)  # Fraction(3, 13)
print(TAU)           # Fraction(3472, 891)
print(KAPPA_T)       # Fraction(1, 61)

# v1.1.0: 12 additional topological relations
print(GAMMA_GIFT)    # Fraction(511, 884)
print(THETA_23)      # Fraction(85, 99)
print(ALPHA_INV_BASE)  # 137

# v1.3.0: Yukawa duality relations
print(ALPHA_SUM_A)   # 12 (Structure A: 2+3+7)
print(ALPHA_SUM_B)   # 13 (Structure B: 2+5+6)
print(ALPHA_PROD_A + 1)  # 43 (visible sector)
print(ALPHA_PROD_B + 1)  # 61 (kappa_T inverse)
print(DUALITY_GAP)   # 18 (colored sector correction)

# v1.4.0: Irrational sector (exact rationals!)
print(ALPHA_INV_COMPLETE)  # Fraction(267489, 1952) ~ 137.033
print(THETA_13_DEGREES_SIMPLIFIED)  # Fraction(60, 7) ~ 8.571 degrees
print(PHI_LOWER_BOUND)  # Fraction(1618, 1000) = 1.618
print(M_MU_M_E_LOWER)   # 206 < 27^phi < 208

# All proven relations
for r in PROVEN_RELATIONS:
    print(f"{r.symbol} = {r.value}")

K7 Metric Pipeline (v1.2.0)

Build your own G2 holonomy metrics on K7 manifolds:

import gift_core as gc

# Check if numpy is available (required for K7 modules)
if gc.NUMPY_AVAILABLE:
    # Configure the pipeline
    config = gc.PipelineConfig(
        neck_length=15.0,      # TCS gluing parameter
        resolution=32,         # Grid resolution
        pinn_epochs=1000,      # Neural network training
        use_pinn=True          # Enable physics-informed learning
    )

    # Run the full G2 geometry pipeline
    result = gc.run_pipeline(config)

    # Access computed values
    print(f"det(g) = {result.det_g}")      # Should match 65/32
    print(f"kappa_T = {result.kappa_T}")    # Should match 1/61
    print(f"b2 = {result.betti[2]}")        # 21
    print(f"b3 = {result.betti[3]}")        # 77

    # Export certificate to Lean/Coq
    lean_proof = result.certificate.to_lean()
    coq_proof = result.certificate.to_coq()

    # Physics extraction
    yukawa = gc.YukawaTensor(result.harmonic_forms)
    masses = yukawa.fermion_masses()

Requirements: K7 modules require numpy. Install with:

pip install giftpy numpy

---

Proven Relations (39 Total)

All relations are formally verified in both Lean 4 and Coq. Each relation is a mathematical identity; no fitting or approximation is involved.

Original 13 Relations

#	Symbol	Value	Derivation	Status
1	sin²θ_W	3/13	b₂/(b₃ + dim(G₂)) = 21/91	Lean + Coq
2	τ	3472/891	dim(E₈×E₈)·b₂ / (dim(J₃O)·H*)	Lean + Coq
3	κ_T	1/61	1/(b₃ - dim(G₂) - p₂)	Lean + Coq
4	det(g)	65/32	(H* - b₂ - 13) / 2^(Weyl)	Lean + Coq
5	Q_Koide	2/3	dim(G₂)/b₂ = 14/21	Lean + Coq
6	m_τ/m_e	3477	dim(K₇) + 10·dim(E₈) + 10·H*	Lean + Coq
7	m_s/m_d	20	p₂² × Weyl = 4 × 5	Lean + Coq
8	δ_CP	197°	dim(K₇)·dim(G₂) + H*	Lean + Coq
9	λ_H (num)	17	H* - b₂ - 61	Lean + Coq
10	H*	99	b₂ + b₃ + 1	Lean + Coq
11	p₂	2	dim(G₂)/dim(K₇)	Lean + Coq
12	N_gen	3	rank(E₈) - Weyl	Lean + Coq
13	dim(E₈×E₈)	496	2 × 248	Lean + Coq

Topological Extension (12 New Relations) - v1.1.0

#	Symbol	Value	Derivation	Status
14	α_s denom	12	dim(G₂) - p₂	Lean + Coq
15	γ_GIFT	511/884	(2·rank(E₈) + 5·H*) / (10·dim(G₂) + 3·dim(E₈))	Lean + Coq
16	δ penta	25	Weyl² (pentagonal structure)	Lean + Coq
17	θ₂₃	85/99	(rank(E₈) + b₃) / H*	Lean + Coq
18	θ₁₃ denom	21	b₂ (Betti number)	Lean + Coq
19	α_s² denom	144	(dim(G₂) - p₂)²	Lean + Coq
20	λ_H²	17/1024	(dim(G₂) + N_gen) / 32²	Lean + Coq
21	θ₁₂ factor	12775	Weyl² × γ_num	Lean + Coq
22	m_μ/m_e base	27	dim(J₃(O))	Lean + Coq
23	n_s indices	11, 5	D_bulk, Weyl_factor	Lean + Coq
24	Ω_DE frac	98/99	(H* - 1) / H*	Lean + Coq
25	α⁻¹ base	137	(dim(E₈) + rank(E₈))/2 + H*/11	Lean + Coq

Yukawa Duality (10 New Relations) - v1.3.0

The Extended Koide formula exhibits a duality between two α² structures that are both topologically determined:

Structure	α² values	Sum	Product+1	Physical meaning
A (Topological)	{2, 3, 7}	12 = gauge_dim	43 = visible	K3 signature origin
B (Dynamical)	{2, 5, 6}	13 = rank+Weyl	61 = κ_T⁻¹	Exact mass fit

The torsion κ_T = 1/61 mediates between topology and physics.

#	Symbol	Value	Derivation	Status
26	α²_A sum	12	2 + 3 + 7 = dim(SM gauge)	Lean + Coq
27	α²_A prod+1	43	2×3×7 + 1 = visible_dim	Lean + Coq
28	α²_B sum	13	2 + 5 + 6 = rank(E₈) + Weyl	Lean + Coq
29	α²_B prod+1	61	2×5×6 + 1 = κ_T⁻¹	Lean + Coq
30	Duality gap	18	61 - 43 = p₂ × N_gen²	Lean + Coq
31	α²_up (B)	5	dim(K₇) - p₂ = Weyl	Lean + Coq
32	α²_down (B)	6	dim(G₂) - rank(E₈) = 2×N_gen	Lean + Coq
33	visible_dim	43	b₃ - hidden_dim	Lean + Coq
34	hidden_dim	34	b₃ - visible_dim	Lean + Coq
35	Jordan gap	27	61 - 34 = dim(J₃(𝕆))	Lean + Coq

Irrational Sector (4 New Relations) - v1.4.0

Relations involving irrational numbers (pi, phi) with certified rational parts:

#	Symbol	Value	Derivation	Status
36	alpha^-1 complete	267489/1952	128 + 9 + (65/32)(1/61)	Lean + Coq
37	theta_13 degrees	60/7	180/b2 = 180/21	Lean + Coq
38	phi bounds	(1.618, 1.619)	sqrt(5) in (2.236, 2.237)	Lean + Coq
39	m_mu/m_e bounds	(206, 208)	27^phi	Lean + Coq

Key insight: The fine structure constant inverse alpha^-1 = 267489/1952 is an exact rational, not an approximation! This comes from:

• 128 = (dim(E8) + rank(E8))/2 = algebraic component
• 9 = H*/D_bulk = bulk component
• 65/1952 = det(g) * kappa_T = torsion correction

Topological Constants

The relations above derive from these fixed mathematical structures:

Constant	Value	Origin
b₂(K₇)	21	Second Betti number of K₇ manifold
b₃(K₇)	77	Third Betti number of K₇ manifold
dim(G₂)	14	Dimension of G₂ holonomy group
dim(K₇)	7	Dimension of internal manifold
rank(E₈)	8	Rank of E₈ Lie algebra
dim(E₈)	248	Dimension of E₈ Lie algebra
dim(J₃O)	27	Dimension of exceptional Jordan algebra
Weyl	5	Weyl factor from E₈ structure
D_bulk	11	M-theory bulk dimension

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Formal Verification

Why Two Proof Assistants?

Independent verification in Lean 4 and Coq provides strong confidence that these mathematical identities are correct. The proofs verify:

1. Arithmetic correctness: All rational simplifications (e.g., 21/91 = 3/13) are exact.
2. Algebraic consistency: Relations using shared constants are mutually compatible.
3. Integer constraints: Values claimed to be integers are proven integral.

What Formal Verification Does NOT Prove

The proofs establish mathematical identities only. They do not verify:

• That these formulas correspond to physical observables
• That the topological interpretation is correct
• That the underlying physics framework is valid

Building the Proofs

# Lean 4
cd Lean && lake build

# Coq
cd COQ && make

Proof Structure

Lean/
├── GIFT/
│   ├── Algebra.lean           # E8, G2 structures
│   ├── Topology.lean          # Betti numbers
│   ├── Geometry.lean          # K7, Jordan algebra
│   ├── Relations.lean         # Original 13 relations
│   ├── Relations/
│   │   ├── GaugeSector.lean   # α_s, α⁻¹ structure
│   │   ├── NeutrinoSector.lean # θ₁₂, θ₁₃, θ₂₃, γ_GIFT
│   │   ├── LeptonSector.lean  # m_μ/m_e, λ_H²
│   │   ├── Cosmology.lean     # n_s, Ω_DE
│   │   ├── YukawaDuality.lean # α² duality (v1.3.0)
│   │   ├── IrrationalSector.lean  # θ₁₃, θ₂₃, α⁻¹ (v1.4.0)
│   │   └── GoldenRatio.lean   # φ, m_μ/m_e (v1.4.0)
│   └── Certificate.lean       # Master theorem (39 relations)
└── lakefile.lean

COQ/
├── Algebra/                   # E8, G2 structures
├── Topology/                  # Betti numbers
├── Geometry/                  # K7, Jordan algebra
├── Relations/                 # All relation files
│   ├── Weinberg.v
│   ├── Physical.v
│   ├── GaugeSector.v
│   ├── NeutrinoSector.v
│   ├── LeptonSector.v
│   ├── Cosmology.v
│   ├── YukawaDuality.v        # v1.3.0
│   ├── IrrationalSector.v     # v1.4.0
│   └── GoldenRatio.v          # v1.4.0
└── Certificate/
    └── AllProven.v            # Master theorem (39 relations)

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K7 Metric Modules (v1.2.0)

The K7 pipeline enables computation of G2 holonomy metrics on twisted connected sum (TCS) manifolds.

Module Architecture

gift_core/
├── geometry/          # Manifold construction
│   ├── kummer_k3.py   # Kummer K3 surface (T^4/Z_2 resolution)
│   ├── acyl_cy3.py    # Asymptotically cylindrical CY3
│   ├── tcs_construction.py  # TCS gluing (X+ ∪ X-)
│   └── k7_metric.py   # Full K7 metric assembly
│
├── g2/                # G2 geometry
│   ├── g2_form.py     # 3-form φ with 35 components
│   ├── holonomy.py    # Holonomy group computation
│   ├── torsion.py     # Torsion-free conditions (dφ=0, d*φ=0)
│   └── constraints.py # GIFT constraints (det_g, κ_T)
│
├── harmonic/          # Hodge theory
│   ├── hodge_laplacian.py   # Δ = dd* + d*d
│   ├── harmonic_forms.py    # Kernel extraction
│   └── betti_validator.py   # Validate b2=21, b3=77
│
├── nn/                # Machine learning
│   ├── fourier_features.py  # Positional encoding
│   └── g2_pinn.py     # Physics-informed neural network
│
├── physics/           # Observable extraction
│   ├── yukawa.py      # Yukawa tensor from harmonic 3-forms
│   ├── mass_spectrum.py     # Fermion mass hierarchies
│   └── gauge_couplings.py   # sin²θ_W, α_s from geometry
│
├── verification/      # Certified computation
│   ├── interval.py    # Interval arithmetic bounds
│   ├── certificate.py # G2Certificate generation
│   └── lean_export.py # Export proofs to Lean/Coq
│
└── pipeline/          # End-to-end workflow
    └── pipeline.py    # GIFTPipeline orchestration

Key Classes

Class	Purpose
KummerK3	T^4/Z_2 resolution with 16 blown-up singularities
ACylCY3	Asymptotically cylindrical Calabi-Yau 3-fold
TCSManifold	Twisted connected sum K7 = X+ ∪_φ X-
G2Form	Associative 3-form with 35 independent components
HodgeLaplacian	Discrete Laplacian for harmonic form extraction
G2PINN	Neural network satisfying G2 torsion constraints
YukawaTensor	Yukawa couplings from triple harmonic integrals
G2Certificate	Certified bounds exportable to Lean/Coq
GIFTPipeline	Full end-to-end metric computation

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Experimental Comparison

For context, these exact values may be compared to experimental measurements. Discrepancies, if any, could indicate either experimental uncertainty or limitations of the conjectured physical correspondence.

Observable	GIFT (exact)	Experimental (PDG 2024)	Deviation
sin²θ_W	0.230769...	0.23122 ± 0.00004	0.20%
m_s/m_d	20	20.0 ± 1.0	0.00%
m_τ/m_e	3477	3477.0 ± 0.1	0.00%
Q_Koide	0.666...	0.666661 ± 0.000007	0.001%
δ_CP	197°	197° ± 24°	0.00%

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Scope and Limitations

What This Package Provides

• 39 exact relations with formal proofs (13 original + 12 extension + 10 Yukawa + 4 irrational)
• Python interface for numerical computation
• Cross-verified mathematical identities
• K7 metric pipeline (v1.2.0): Build G2 holonomy metrics from scratch
• Yukawa duality (v1.3.0): Topological ↔ dynamical α² structure
• Irrational sector (v1.4.0): Exact alpha^-1, golden ratio bounds

What This Package Does Not Claim

• Physical validity of the GIFT framework
• Correctness of the E₈×E₈ / G₂ holonomy interpretation
• Agreement beyond current experimental precision
• Resolution of open problems in theoretical physics

Falsifiability

The framework makes testable predictions. Key experimental tests include:

• DUNE (2027+): δ_CP = 197° ± 10° would test the CP violation prediction
• FCC-ee: High-precision sin²θ_W measurements could distinguish 3/13 from experiment
• Lattice QCD: Improved m_s/m_d determinations test the exact integer prediction

Significant deviation from these predictions would challenge the physical interpretation.

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Related Resources

Full Framework

• Repository: gift-framework/GIFT

Citation

If referencing this work:

@software{gift_core,
  author = {deLaFournière, Brieuc},
  title = {GIFT Core: Formally Verified Mathematical Constants},
  year = {2025},
  url = {https://github.com/gift-framework/core},
  note = {Lean 4 and Coq verified, 39 certified relations}
}

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License

MIT License

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Acknowledgments

This work benefited from computational assistance provided by various AI systems used as mathematical tools. The formal verification was developed using Lean 4 and Coq proof assistants.

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GIFT Core v1.4.0 - 39 Certified Relations + K7 Metric Pipeline + Yukawa Duality + Irrational Sector