We study an infinite-horizon discounted mean field control problem for finite-state nonlinear continuous-time Markov chains. Controls are locally integrable transition-rate matrices and are penalized by relative entropy with respect to a state-dependent reference generator. We prove that the value function is the unique bounded viscosity solution on the closed simplex of the Hamilton–Jacobi–Bellman equation by a direct dynamic-programming argument and a finite-state comparison principle. Under a uniform semiconvexity condition and C¹,¹ regularity, the value function is globally semiconvex and intrinsically C¹,¹ on the closed simplex and yields a Gibbs feedback that is the unique Hamiltonian minimizer at interior states. For positive, symmetric, possibly state-dependent reference rates, an entropy-plus-value free energy decays globally exponentially when the temperature parameter is compatible with the semiconvexity modulus. We also establish a discounted planner-to-game correspondence for potential mean field games under a state-independent reference generator.