Collatz Sequence Dynamics: Phenomenological Patterns, Modular Structures, and Lower-Digit Dominance (Japanese Version)

This study explores the phenomenological behavior of Collatz sequences, focusing on modular transitions and the dominance of lower-digit structures. By examining initial values constructed from sequential prime products, as well as random large odd numbers, we observe a consistent pattern where the maximum value in the sequence tends to emerge shortly after a transition chain of the form 8m+6 → 8m+2 → 8m. Notably, these maxima overwhelmingly align with even numbers of the 8m type.

A key hypothesis proposed is that the behavior of the Collatz sequence is largely governed by the lower 5 or 6 digits of the initial value, regardless of its overall magnitude. This "lower-digit dominance" suggests that sequences with identical lower digits follow similar trajectories—even when their upper digits differ significantly.

Through computational experiments and modular analysis, this work highlights unexpected regularities and structural motifs in an otherwise chaotic landscape. These findings open new avenues for classifying Collatz behavior using modular patterns and data-driven clustering.

本研究は Collatz 系列における現象論的構造を検討し、特にモジュラー遷移と初期値における下位桁の支配性に注目する。8m+6 → 8m+2 → 8m の遷移鎖を経て系列内最大値が現れる傾向を観察し、これらが高確率で 8m 型の偶数であることを確認した。

主要な仮説として、系列の軌道は初期値の下5〜6桁によって大きく影響を受ける「下位桁支配性」が提起され、これは構造的分類の新しい可能性を示すものである。

数値実験とモジュラー解析によって、この混沌とした力学系に潜む規則性を明らかにし、軌道構造の分類やクラスタリングに応用できる視座を提供する。

This preprint complements a previous English version submitted in June 2025