Jones index of prime endomorphisms in the Bost–Connes system: higher invariants, fusion rules, and logarithmic corrections to black hole entropy

We prove that the Jones index of the prime endomorphisms ρ_p in the Bost–Connes system equals p, provide complete proofs via the Takesaki crossed product, and compute the higher invariants: abelian fusion rules (ρ_n ∘ ρ_m = ρ_nm), one-dimensional intertwiner spaces, and group-type Ocneanu invariants for p=2 (ℤ/2ℤ) and p=3 (ℤ/3ℤ). The Riemann zeta function is expressed as ζ(β) = ∏_p (1 − [M:N]_p−β)⁻¹. We connect the Bost–Connes subfactors with CFT₂ minimal models via the ADE classification: BC and Ising share Jones index [M:N]=2 but have inequivalent fusion categories. Motivated by Wang and He (arXiv:2603.07639, 2026), who demonstrated a quantitative BTZ dual for the critical Ising chain, we predict k_Ising ≠ 3/2 for the logarithmic correction coefficient to BTZ entropy at c=1/2.