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scratchpad7

-              v8
+              v9
 Also note, that the indexing of the temperatures is commutative, and that all of the diagonal temperature terms will have two contributions.  One from $\alpha_{\pm i \pm j}$ and one from $\alpha_{\pm j \pm i}$.  We can rewrite
+$\displaystyle \sum_{\pm i, \pm j,i \ne j} \alpha_{\pm i, \pm j} T^{\lambda+1}_{\pm \hat{i} \pm \hat{j}} = \sum_{\pm i, \pm j,i < j} \left ( \alpha_{\pm i, \pm j} + \alpha_{\pm j, \pm i} \right ) T^{\lambda+1}_{\pm \hat{i} \pm \hat{j}} = \sum_{\pm i, \pm j,i < j} \alpha*_{\pm i, \pm j}T^{\lambda+1}_{\pm \hat{i} \pm \hat{j}}$
+$\displaystyle \sum_{\pm i, \pm j,i \ne j} \alpha_{\pm i, \pm j} T^{\lambda+1}_{\pm \hat{i} \pm \hat{j}} = \sum_{\pm i, \pm j,i < j} \left ( \alpha_{\pm i, \pm j} + \alpha_{\pm j, \pm i} \right ) T^{\lambda+1}_{\pm \hat{i} \pm \hat{j}} $
+$= \sum_{\pm i, \pm j,i < j} \alpha*_{\pm i, \pm j}T^{\lambda+1}_{\pm \hat{i} \pm \hat{j}}$
 where