Jones index of prime endomorphisms in the Bost–Connes system

We compute the Jones index of the inclusion ρ_p(R) ⊂ e_p R e_p in the Bost–Connes system, obtaining [e_p R e_p : ρ_p(R)] = p for each prime p. This yields an operator-algebraic interpretation of the Euler product for the Riemann zeta function. We establish that the partially ordered set of subfactors {ρ_n(R)} is isomorphic to the divisibility lattice of the natural numbers with reverse order, with prime subfactors mutually perpendicular in the sense of Pimsner–Popa. The Longo–Witten entropy bound gives ΔS ≤ log p per prime, decomposing additively over prime factors for composite indices. These results, while derivable from known ingredients (Laca–Neshveyev 2004, Longo 1989, Connes–Marcolli 2008), do not appear as explicit statements in the published literature.