Relevant alternative analytic average magnetization calculation method for the square and the honeycomb Ising lattices

In this work, the order parameter or average magnetization expressions are obtained for the square and the honeycomb lattices based on recently obtained magnetization relation, $<\sigma_{0,i}>= <\!\!\tanh[ \kappa(\sigma_{1,i}+\sigma_{2,i}+\dots +\sigma_{z,i})+H]\!\!>$. Where, $\kappa$ is the coupling strength and $z$ is the number of nearest neighbors. $\sigma_{0,i}$ denotes the central spin at the $i^{th}$ site while $\sigma_{l,i}$, $l=1,2,\dots,z$, are the nearest neighbor spins around the central spin. In our investigation, inevitably we have to make a conjecture about the three site correlation function appearing in the obtained relation of this paper. The conjectured form of the the three spin correlation function is given by the relation, $<\!\!\sigma_{1}\sigma_{2}\sigma_{3}\!\!>=a<\sigma>+(1-a)<\sigma>^{(1+\beta^{-1})}$, here $\beta$ denotes the critical exponent for the average magnetization and $a$ is positive real number less than one. The relevance of this conjecture is based on fundamental physical reasoning. In addition, it is tested and investigated by comparing the obtained relations of this paper with the previously obtained exact results for the square and honeycomb lattices. It is seen that the agreements of the obtained average magnetization relations with those of the previously obtained exact results are unprecedentedly perfect.