6.4 Making the correct choice for diagonal adjacency.

So we must choose an adjacency rule that makes $\bf {X}_{-f}^{> - \tau} = \bf {X}_{f}^{< \tau}$. Now the sets $\bf {X}_{f}^{\geq \tau}$ obviously satisfy monotonicity as $\tau$ is decreased, in the sense that once a point becomes a member of the set it remains a member. This is obviously true as $f(p) \geq c$ implies $f(p) > \tau$ for $\tau < c$. This implies that if two Highs of a 4-hit face F are adjacent for a given threshold c, then they must then be adjacent for all values $\tau < c$. For 3 dimensions and higher we have to decide at what value there is a path through the cube interior between the two Highs. The different choices effect the values of certain types of criticalities but will not change the essential character of our results.