Project T Volume XXXIII : The φ² Mixing Theorem Spectral Analysis of S₅ in Q(√2, √5) and the Golden Mixing Time

We establish that the mixing time of the Cayley graph of S5 with adjacent transpositions equals φ² = (3 + √5)/2, where φ is the golden ratio. This follows from the spectral gap λ1(S5) = 2 − φ, whose reciprocal is algebraically identical to φ². We classify the complete spectrum of the S5 Cayley graph Laplacian: 25 distinct eigenvalues organized into four families—integer (5 levels, 26 modes), √2 (4 levels, 20 modes), φ (10 levels, 44 modes), and an extended family (6 levels, 30 modes)—with all eigenvalues living in Q(√2, √5). The multiplicity sequence [1, 4, 5, 4, 5, 6, 5, 4, 6, 5, 5, 4, 12, 4, 5, 5, 6, 4, 5, 6, 5, 4, 5, 4, 1] is a perfect palindrome centered at λ = 4. We develop the spectral zeta function ζL(s) across the chain S3 ⊂ S4 ⊂ S5 and establish structural scaling relations. These results complement Volume XXXII’s Mass Ladder Theorem and prepare Volume XXXIV’s renormalization analysis.