PROJECT T Volume XXVI : Moonshine from the Spectrum How the Complex Eigenvalues of H₀(Ã₂) Map to the Zero of the j-Invariant

We establish an exact algebraic bridge between the spectral theory of the 0-Hecke monoid H₀(Ã₂) and the modular invariant j(τ). Our central result is:
Theorem. j(λ₊/K) = 0, where λ₊ = (3+i√3)/2 is the complex eigenvalue of the transfer operator T = πₐ + πᵇ + πᶜ on ℂ[S₃], and K = 3 is the number of generators.
The proof is purely algebraic: the characteristic polynomial λ² – 3λ + 3 = 0 forces λ₊ = 2 + ω, an Eisenstein integer of norm K = 3. The normalized eigenvalue τ = λ₊/K lives in the upper half-plane and is SL(2,ℤ)-equivalent to ω = e^{2iπ/3}, the ℤ₃ fixed point where j vanishes. Moreover, K = 3 is the unique positive integer for which this j-zero condition holds.
This establishes a triple selection principle for K = 3: (1) C(K) = K(K−3)/2 = 0 (commutation deficit), (2) j(λ₊/K) = 0 (modular zero), (3) disc(λ²–Kλ+K) = −3 (Eisenstein discriminant). The results connect the three levels of moonshine—Mathieu (c = 6), Conway (c = 12), and Monster (c = 24)—to the algebraic data of H₀(Ã₂) and H₀(A₃).