ALGEBRAIC RESOLUTION Foundation III : The Spectrum Complex eigenvalues, semisimplicity, and ℤ₃ symmetry

This volume completes the spectral analysis of the affine 0-Hecke monoid H₀(Ã₂) acting on the group algebra ℂ[S₃]. Where Foundation I established the uniqueness of K = 3 through the commutation deficit C(K) = K(K−3)/2, and Foundation II revealed the Pell crystal structure governing the bulk statistics, Foundation III confronts the full eigenvalue problem of the transfer operator T = T₁ + T₂ + T₀. The central discovery is that T possesses a complex spectrum: its characteristic polynomial factorizes as λ(λ−3)(λ²−3λ+3)², yielding a conjugate pair of complex eigenvalues λ± = (3 ± i√3)/2. These eigenvalues encode the rank through their modulus (|λ±|² = K = 3) and the hexagonal geometry of Ã₂ through their argument (arg(λ±) = π/6 = 30°). Every result in this volume is computed in exact integer arithmetic and cross-validated against the computational script foundation_iii_spectrum.py. All 19 verification checks pass without exception. Zero free parameters are introduced at any stage.