Lean Formalization

Overview

The K₇ framework is formally verified in Lean 4 with Mathlib. The formalization establishes that all claimed algebraic relations between topological inputs and physical predictions follow by pure computation.

Metric	Value
Lean files	143
Build jobs	8391
Axioms	15 (4 main-chain + 11 K3 interval-arithmetic)
sorry statements	0
Warnings	0
Certificate conjuncts	140 (across the Foundations / Predictions / Spectral pillars)

Repository

Code: github.com/Arithmon/K7-Lean Blueprint: arithmon.github.io/K7-Lean Lean version: 4.29.0 Mathlib version: 4.29.0

Architecture

K7-Lean/Lean/K₇/
├── Core.lean              # Constants (dim_E8, b2, b3, H*, ...)
├── Certificate.lean       # Master theorem (127 conjuncts)
├── Foundations/            # E8 roots, G2 cross product
├── Geometry/               # Differential geometry
├── Spectral/               # Spectral theory, mass gap, Yukawa
├── Relations/              # Physical predictions
│   ├── GaugeSector.lean
│   ├── NeutrinoSector.lean
│   ├── LeptonSector.lean
│   ├── YukawaDuality.lean
│   └── ...
├── ExplicitG2Metric.lean   # 400 lines
├── NewtonKantorovich.lean   # 401 lines, 0 axioms
├── K3Harmonic.lean          # 447 lines
├── K7Orthonormality.lean    # 0 axioms, 13 theorems
├── TCSGaugeBreaking.lean    # 0 axioms, 14 theorems
├── GaugeBundleData.lean     # 0 axioms, 12 theorems
├── AssociativeVolumes.lean  # 0 axioms, 19 theorems
├── CompactificationCorrection.lean  # δ_CP = 12214/69
└── ComputedWeylLaw.lean     # 0 axioms, 8 theorems

Key Certificates

Master Certificate (127 conjuncts)

The K₇_framework_certified theorem verifies all relations in one compile:

Foundations (34): E₈ dimensions, G₂ structure, K₇ topology, Weyl group
Predictions (56): Gauge, lepton, quark, neutrino, CKM, boson, cosmological
Spectral (37): Mass gap, Yukawa ratios, Weyl law, orthonormality

Zero-Axiom Modules

Several modules prove their results with zero domain-specific axioms:

Module	Theorems	Certificate
NewtonKantorovich	,	NK convergence
TCSGaugeBreaking	14	10 conjuncts
GaugeBundleData	12	11 conjuncts
AssociativeVolumes	19	14 conjuncts
CompactificationCorrection	12	6 conjuncts
ComputedWeylLaw	8	7 conjuncts
K7Orthonormality	13	Gram matrices

Axiom Audit

All 7 axioms are substantive (standard mathematical theorems or K₇ conjectures). Zero are placeholders. Categories:

Topological inputs: b₂ = 21, b₃ = 77, dim(G₂) = 14, etc.
E₈ properties: Root system, Weyl group factorization
Joyce/anistropic existence input: compact torsion-free G_2 existence on K_7 is an analytic assumption/target, tracked in docs/analytic_status.md
Spectral bounds: Numerical results from certified computation

Verification

To verify all proofs locally:

git clone https://github.com/Arithmon/K7-Lean
cd K7-Lean/Lean
lake build

Completes in ~30 seconds (warm cache). CI rebuilds on every commit.

What is Proven vs Not Proven

Proven: Given topological inputs (b₂, b₃, dim(G₂), etc.), all 127 algebraic relations follow by pure computation. Zero domain-specific axioms for the arithmetic.

Not proven: (1) Existence of K₇ with these specific Betti numbers is axiomatized, not constructed from scratch. (2) Physical interpretation of the relations. (3) Uniqueness of the construction.

For Formalization Experts: Methodology and context
Paper Main Framework: Physics predictions these proofs certify
Paper S2 Derivations: The derivations being formalized