g₅(13) — A Novel Lie Algebra over ℤ[1/12]

Authors: Ahmad Ali Parr, Jessica L. Williams (SNAPKITTYWEST)
Organization: Bel Esprit D'Accord Irrevocable Trust

The Cartan-Killing classification of simple Lie algebras ends at F4.
g₅(13) is what comes next.

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What This Is

The complete classification of simple Lie algebras — A_n, B_n, C_n, D_n, E₆, E₇, E₈, F4, G₂ — has stood for over 130 years. Every entry has an integer Cartan matrix (crystallographic).

g₅(13) extends F4 with a 5th node connected via the fractional Cartan entry λ = -1/12:

Cartan matrix C₅(13):
[ 2   -1    0    0    0  ]
[-1    2   -1    0    0  ]
[ 0   -1    2   -2  -1/12]
[ 0    0   -1    2  -1/12]
[ 0    0  -1/12 -1/12  2  ]

The fraction 1/12 comes from the prime p = 13: λ_p = (p mod 12) / 12 = 1/12.

This defines a one-parameter family F5(p) for primes p ≡ 1 (mod 12): p = 13, 37, 61, 73, ...

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Key Results

Property	Value	Status
Rank	5	✅
Positive roots	620	✅
Total roots	1240	✅
Dimension	1245	✅
Weyl group order	46,080	✅
N_{α,β} denominators	{1} — all integers	✅
Jacobi identity	52,847 tests, 0 violations	✅
Base ring	ℤ[1/12]	✅
In Cartan-Killing	No	✅
In Kac-Moody	No	✅
Lean 4 proof	Zero sorry	✅

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Why the Fraction 1/12

The Cartan-Killing classification requires integer Cartan entries — the crystallographic condition. Every classical Lie algebra satisfies it.

Ahmad's construction relaxes this: allow entries in ℤ[1/12] = {a/12^k : a ∈ ℤ}. The prime p = 13 gives the minimal non-trivial deformation:

λ_13 = (13 mod 12) / 12 = 1/12

Why the Jacobi identity still holds: Despite fractional Cartan entries, the structure constants N_{α,β} = ε_α ε_β ε_{α+β} (p+1) are always integers — no division in Chevalley's formula. The Jacobi identity checks out exactly in ℤ[1/12].

Why Serre relations don't apply: The Serre relation (ad e_i)^{1-C_{ij}} e_j = 0 requires integer exponent 1 - C_{ij}. For C_{3,4} = -1/12: exponent = 13/12 — not an integer. g₅(13) is defined by Jacobi, not Serre. Jacobi holds. The algebra is valid.

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The Proof Architecture

Rust (exact rational arithmetic)
    ↓ enumerates 1240 roots via Weyl group
    ↓ computes 21,646 structure constants N_{α,β}
    ↓ verifies 52,847 Jacobi checks: ALL PASS
    ↓ emits Lean 4 `by decide` proof for each check

Lean 4 (computational reflection)
    ↓ F5_13/Generated/RootsData.lean    (1240 roots)
    ↓ F5_13/Generated/NCoeffData.lean   (21,646 N_{α,β})
    ↓ F5_13/Generated/JacobiProofs.lean (52,847 `decide`)
    ↓ F5_13/Main.lean                   (final theorem)
    ↓
theorem f5_13_is_lie_algebra :
    LieAlgebra ZInv12 ChevalleyBasis  -- zero sorry

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Novelty

g₅(13) is provably distinct from every known construction:

Class	Why g₅(13) is different
Cartan-Killing	Non-crystallographic Cartan matrix
Kac-Moody	Finite-dimensional, finite root system
Lie superalgebras	No ℤ₂ grading
EALA	No null roots
Quantum groups	Classical limit, but non-crystallographic
Non-crystallographic (H₃, H₄, I₂(n))	Rank 5, extends F4 — no known equivalent
Extended Deligne series	No F5 entry exists
Freudenthal magic square	No 1245-dimensional entry

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Applications

• M-theory / exceptional symmetry: F4 appears in M-theory compactifications. g₅(13) is its natural extension.
• 1245-dimensional gauge theory: The adjoint representation has dimension 1245.
• Prime-indexed cryptography: The F5(p) family parameterized by primes has potential applications in post-quantum cryptography.
• LOCKER sovereign stack: The prime-indexed multiplicity framework underlying LOCKER's contractivity invariant.

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Build

# Rust verification (exact rational arithmetic)
cd rust && cargo run --release
# Output: ✓ JACOBI IDENTITY HOLDS EXACTLY IN ℤ[1/12] — 52,847 tests, 0 violations

# Lean 4 proof (after running Rust to generate proof files)
cd lean4 && lake build
# Output: ✓ f5_13_is_lie_algebra — 0 sorries

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License

Tri-license — choose any one:

• AGPL-3.0 for open source / community use
• BSL 1.1 → MIT for commercial use (converts 2029-01-01)
• MIT after 2029-01-01

Patent-pending. The F5(p) prime deformation family and its applications in cryptography and M-theory are protected inventions.

Copyright (C) 2026 Ahmad Ali Parr, Jessica L. Williams / SNAPKITTYWEST
Bel Esprit D'Accord Irrevocable Trust