S = ∫M⁴×X⁸ d¹²x √−g₁₂ [
M₁₂¹⁰/2 · R₁₂ −
1/(4g₁₂²) Tr FMN FMN +
Ψ̄ iΓMDMΨ ]
M, N = 0, 1, …, 11 · X⁸ = T⁸/W(E8) the Coxeter torus orbifold · FMN ∈ 𝔢₈ (adjoint, dim 248) · Ψ ∈ 248 (adjoint fermion)
Term 1 · Gravity
(M₁₂¹⁰/2) R₁₂
12D Einstein-Hilbert
Dimensional reduction on T⁸ yields 4D gravity. The spectral gap λ₁ = 8π² of the E8 lattice Laplacian generates the exponential hierarchy. Mixed-index components gμm become gauge fields via Kaluza-Klein. Internal scalars gmn contain the Higgs through gauge-Higgs unification.
What it produces
Planck mass
MPl = v · e4π²/√R 0.38%
Hierarchy
MPl/v ≈ 10¹⁶
Higgs mass
v√R/√d = 125.06 GeV 0.15%
Cosmo. constant
8π²/(Nh)³² ≈ 10⁻¹²² 0.002%
Spectral tilt
ns = 29/30 = 0.9667 0.2%
Tensor-to-scalar
r = 1/900 ≈ 0.001
Term 2 · Gauge
−1/(4g²) Tr FMNFMN
E8 Yang-Mills
FMN is the 248-dim E8 field strength. Wilson lines on the Coxeter torus break E8 → SU(3)×SU(2)×U(1) via the Hosotani mechanism. All couplings determined by Killing form projections. The E8 theta function ΘE8 = E₄ proves ζΔ(0) = −1, giving the 1/e factor.
What it produces
Fine structure
α⁻¹ = 4πh/e · √(N/d) · η 0.003%
Strong coupling
αs = h/2R = 30/256
Weinberg angle
sin²θW = 3/m₄ = 3/13
EW VEV
v = d(1−α) = 246.22 GeV
Strong CP
θQCD = 0 (Λ* = Λ)
Dark energy ΩΛ
h/(h+m₄) = 30/43 1.9%
Term 3 · Matter
Ψ̄ iΓMDMΨ
Adjoint Fermions (12D)
Single 248-dim fermion in 12D. Coxeter fixed points at Zp orbifold singularities (p = 2, 3, 5) localize three chiral generations. Domain wall tunneling between fixed points generates mass hierarchy. E8(−24) provides chirality (bypassing Distler-Garibaldi).
What it produces
3 generations
ω(h) = ω(30) = 3 primes
Top quark
mt = 172.74 GeV 0.003%
12 fermion masses
e−β√p · σ₃ lattice sums
CKM matrix
λ, A, sin 2β derived 0.19%
PMNS angles
θ₁₂, θ₂₃, θ₁₃ derived
Proton radius
rp = 0.8411 fm 0.01%
E8 Yang-Mills
12D · 248 generators
→
Compactify on
T⁸/W(E8)
→
Hosotani breaking
E8 → SM gauge group
→
Zero modes at
Z₂, Z₃, Z₅ fixed pts
→
Standard Model
24/26 parameters