Selection as Orbit, Time as Knot: The Bost–Connes Realization of VID's Modular Substrate

We present the first explicit realization of the Virtual Invasion Dynamics (VID) modular substrate in the Bost–Connes system. At the critical inverse temperature β=1, the unique KMS state generates the hyperfinite type III₁ factor R∞. We prove that nontrivial knot classes [eⁱᵗʰ] ∈ H¹(σ, U(R∞)) exist for σ-fixed directions h, establishing the first explicit nontrivial elements of VID §24. We prove a general adiabatic theorem: entropy production under any automorphism dynamics is identically zero, forcing the time arrow to reside in thermodynamic drift paths. We compute entropy production for commuting and non-commuting perturbations, revealing a trade-off between total asymmetry and local production rate. We formulate the Shuttle Group Conjecture: selection of condensation outcomes is an orbit under group action, with knots marking tearing points on the substrate.