Phase Transitions, Non-Extremality (Reconstruction), and Markov Entropy Rate for the Mixed Spin-$(s,\tfrac12)$ Ising Model on a Cayley Tree of Order Three

We investigate the mixed spin-$(s,\tfrac12)$ Ising model on a Cayley tree of order three ($k=3$), extending the approach of \cite{Akin2024}. For the representative case $s=5$, the associated recursion leads to an 11-dimensional dynamical system, and phase-transition regions are examined via the local stability of the disordered (symmetric) fixed point, detected through the condition $|λ_{\max}|\ge 1$ for the Jacobian matrix. To study extremality (non-reconstruction) of the disordered phase, we represent translation-invariant splitting Gibbs measures by tree-indexed Markov chains and compute the relevant Dobrushin coefficients. At the symmetric fixed point we obtain explicit transition kernels and the induced two-step kernel on the spin-$\tfrac12$ layer; its second eigenvalue $λ_2$ yields a spectral reconstruction test consistent with the Kesten--Stigum condition $3|λ_2|^2>1$. In addition, we introduce the Markov entropy rate as a computable thermodynamic/information-theoretic observable and derive closed-form expressions for this entropy rate at the symmetric fixed point for an arbitrary spin $s$. Numerical illustrations for $s=1,2,\ldots,5$ are provided to compare the entropy-rate behavior with the spectral criteria. Our results connect naturally to themes in information theory and statistical physics and are also relevant to reconstruction problems on trees that appear in biology/phylogenetics.