A Compact Notation for Peculiar Properties Characterizing Integer Tetration

We introduce a compact notation to express the congruence speed (in radix-\(10\)) of an integer tetration base \(a\), along with the cycle of the rightmost non-stable digits of \(^{b}a\) for unit increments of \(b\). The aforementioned discrete function provides a useful tool for efficiently computing the exact number of frozen digits that characterize the right tail of each nontrivial integer tetration. We also provide an improved upper bound for the minimum hyperexponent \(\bar{b}(a)\) that ensures the constancy of the congruence speed of \(a\) for all heights \(b \geq \bar{b}(a)\). Moreover, we prove that the minimum between the constant congruence speeds of any two integers greater than \(1\), whose product is not divisible by \(10\), is always less than or equal to the constant congruence speed of their product. Lastly, we give examples of infinitely many perfect powers whose degree matches their congruence speed at every height above \(2\), emphasizing the peculiar recurrence relations of hyper-\(4\).