Hierarchy mixing threshold conjecture for modular Ackermann maps

For powers-of-two moduli $2^k$, consider the modular Ackermann map $A_{2^k}(m,n)$ and its output distribution on $\mathbb Z_{2^k}$ as the recursion level $m$ varies. Hierarchy mixing threshold conjecture. Based on numerical evidence, level $m=4$ appears to be the first Ackermann level for which $A_{2^k}(m,n)$ exhibits near-uniform distribution over $\mathbb Z_{2^k}$. The experiments indicate a qualitative transition between $m=3$, which retains strong arithmetic structure modulo $2^k$, and $m\ge4$, which exhibits substantially stronger mixing; formal verification remains open.