Towards non-abelian localized Bost-Connes systems: the matrix Kakutani dichotomy and a Tannakian obstruction

For a compact group G with Frobenius classes indexed by a set S of primes, the obstruction series Xi_beta(rho,S) = sum over p of N(p)⁻ᵇᵉᵗᵃ ||I - rho(Frob_p)||_HS^2 defines a Tannakian category T_beta = Rep(G/N_beta) (the matrix Kakutani dichotomy, for commuting lifts), with the Euler dictionary Xi_beta = 2n log zeta_S - 2 Re log L_S + O(1) linking the series to automorphic L-functions (Theorem 4.1). Anomalous sparse sets exist (Theorem 5.3), and Section 6 records why the corresponding non-abelian algebra is not constructed by these methods.