 Solution

For the sawtooth Ising chain, there is a source inconsistency: the printed Ising spin values are $\{-1,r+1\}$, with no definition of $r$. The requested transfer matrix follows for $r=0$, namely the usual Ising model with values $\{-1,+1\}$. For arbitrary $r$, summing an apex spin actually givesFor example, $r=1$, $s=s^{\prime }=-1$ and $\beta \bar{J}=1$ give the bracket $e^{-2}+e^4$, whereas the claimed hyperbolic cosine gives $e^{-2}+e^2$. Thus the printed identity is false literally. The rest of Question 1 uses the intended symmetric Ising spin values, $r=0$.

For a fixed configuration of base spins, each apex spin appears in only one local Boltzmann factor. The apex spins are therefore conditionally independent, and their finite sums factorize:Only the base spins carry information from one triangle to the next. The apex spins have already been summed out, so no apex index remains in the transfer matrix; the final trace implements the periodic boundary condition.