Current research on MMGD is organized around three related questions.

What is preserved when the order of numerical interactions changes?

Multiplication is commutative, but the insertion of a last-nonzero-digit observation after each step can make the resulting finite dynamics order-dependent. The question is which periodic structures remain invariant under changes of order, and under what conditions.

How far does reversal preserve periodic structure, and where does this preservation first fail?

For decimal words of length three, computational results show that reversal preserves cycle type in all 9^3=729 cases. At length four, counterexamples first appear. The current problem is to explain the mechanism behind this boundary.

What can geometric realization distinguish that coarse dynamical invariants cannot?

Different systems may share the same cycle spectrum while differing in their individual orbits or functional graphs. MMGD asks whether the stepwise geometric realization retains and makes visible some of these finer differences.

These questions connect order, finite dynamics, and geometric realization without assuming that they carry the same information.

日本語

現在のMMGD研究は、互いに関連する三つの問いを中心に進めている。

数値相互作用の順序を変えたとき、何が保存されるのか。

乗算そのものは可換である。しかし、各ステップの後に最後の非零桁を選ぶ観測を挿入すると、得られる有限力学は順序に依存しうる。どの周期構造が順序の変更に対して保存され、それがどのような条件で成立するのかを調べている。

反転はどこまで周期構造を保存し、どこで初めて破れるのか。