Project T Volume XXXV : The S₆ Frontier Universal Gap Formula, Number Field Hierarchy, and the Spectral Landscape of 720 Permutations

We compute the complete spectrum of the S6 Cayley graph Laplacian (720 vertices, degree 5, 67 distinct eigenvalue levels) and discover the Universal Gap Formula: λ1(Sn) = 2 − 2cos(π/n). This single expression unifies all previously known spectral gaps: λ1(S3) = 1, λ1(S4) = 2 − √2, λ1(S5) = 2 − φ, and the new result λ1(S6) = 2 − √3. The corresponding mixing times are conjugate surds: τ(S6) = 2 + √3. The full S6 spectrum lives in Q(√2, √3, √5), extending the number field hierarchy where each Sn adds exactly 2cos(π/n) to the algebraic universe. The Mass Ladder Theorem, palindromic degeneracy, and asymptotic freedom all extend to S6, confirming their universality across the symmetric group chain.