Eisenstein Planes, the Leech Lattice, the Cubic Mirror Coordinate, Genus-two Theta Coefficients, and the Euler-Hypergeometric Shadow

Let ω = e^(2πi/3) = (−1 + √(−3))/2, ρ = e^(πi/3) = (1 + √(−3))/2, O = ℤ[ω], L₀ = (2/9)ρ O ⊂ ℂ.

The lattice L₀ is the scaled Eisenstein plane arising in the planar configuration under discussion. We prove that its local equilateral-incircle geometry is the A₂ geometry, that the Niemeier lattice N(12A₂) globalizes twelve such Eisenstein planes, and that the Leech lattice is the holy/Kneser 3-neighbor of N(12A₂). We then prove that the theta series of the Leech lattice yields Klein's j-function by the exact identity

Θ_Λ(τ) = E₄(τ)³ − 720Δ(τ), j(τ) = Θ_Λ(τ)/Δ(τ) + 720.

At the same time, the same order-3 Eisenstein symmetry forces the local descent at the hexagonal CM point ρ to be cubic: every modular invariant has a holomorphic expansion in powers of a local coordinate w only through w³. This gives the precise cause of the cubic mirror coordinate of the Hesse pencil. We then identify the genus-two theta expansion with Fourier coefficients indexed by positive semidefinite binary quadratic forms and show that its boundary degeneration recovers the genus-one theta series and hence the rank-one modular function j. Finally, using the hypergeometric inversion formula for j and the Euler-operator form of the differential equations arising from Ramanujan-Machine type continued fractions, we isolate the common rank-one hypergeometric operator controlling both the hexagonal mirror branch and the Euler branch.