By initially working in the decimal numeral system, we introduce a compact notation to express the congruence speed 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 resulting 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 establish an improved upper bound for the minimum hyperexponent \(\bar{b}(a)\) that guarantees 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. Additionally, still assuming radix-\(10\), we give examples of infinitely many perfect powers whose degree matches their constant congruence speed at every height above \(2\), emphasizing the peculiar recurrence relations of hyper-\(4\). Finally, Appendix~\ref{AppendixB} generalizes the described radix-\(10\) framework to all squarefree numeral systems, showing that only in such systems the congruence speed stabilizes to a fixed (positive) value for all \(a>1\) not divisible by the radical of the radix. Furthermore, we derive compact formulas for all prime-radix numeral systems and for the composite squarefree senary case.

This is Version 7 of the preprint. Compared to Version 1, it adds Appendix B, which generalizes the constant congruence speed to all squarefree numeral systems and provides compact formulas for all prime radices and for the senary case. Previous versions also incorporated several minor improvements in Section 2, including the radix-$r$ generalization of identity (2.12), extending the radix-$10$ result (2.11) to arbitrary $r>1$. The present Version 7 adds the new Remark 7 in Appendix B, which introduces the parameter $\bar{\tau}(r)$, defined as the smallest odd integer greater than $1$ such that the fixed-point equation $y^{\bar{\tau}(r)} = y$ has a maximal set of solutions in the commutative ring of $r$-adic integers $\mathbb{Z}_r$, and gives a compact closed expression for $\bar{\tau}(r)$ in terms of the prime factorization of $r$.