Growth of Mahler measure and algebraic entropy of dynamics with the Laurent property

For the rank-2 case r=2, both the paper and the model prove that the Mahler measure grows linearly, m(x_n)=C* n+O(1), with the same constant C* = ∫_{T^2} |log |(K+√(K^2-4))/2|| dΩ, hence EM(ϕ)=0. The paper establishes the invariant K, linearizes the map to xn+2 − K xn+1 + xn = 0 and derives the slope via the roots e^{±iv} of the characteristic polynomial, giving exactly the same integral formula (Theorem 2.2) . The model follows an equivalent linearization (xn = u λ^n + v λ^{-n}) and supplies a more explicit uniform L^1 control of the O(1) remainder. Aside from this added detail and a minor slip about the dimension of the K=±2 locus on T^2, the arguments are essentially the same.