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 ρₚ in the Bost–Connes system equals p, provide complete proofs via the Takesaki crossed product, and compute the higher invariants: abelian fusion rules (ρₙ ∘ ρₘ = ρₙm), 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: Nₚ^−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 kIsing ≠ 3/2 for the logarithmic correction coefficient to BTZ entropy at c=1/2.