Collatz Reduction via 8m+4 Descent Hypothesis

We present a structural reduction of the Collatz problem based on a systematic analysis of even integers under the Collatz map. The key steps are as follows: 1. We define the Collatz map T(n) as: T(n) = n/2 if n is even, and (3n+1)/2 if n is odd. 2. We introduce the OK Condition: an even integer N is OK if its Collatz orbit contains an even integer M such that 2≦ M ＜ N. This reduces the global convergence problem to a finite descent problem. 3. We classify all even integers modulo 8: - N ≡ 0, 2, 4 (mod 8) satisfy the OK condition immediately. - Only N ≡ 6 (mod 8) may fail to descend. These form the unresolved class N = 8m + 6. 4. Every such N can be uniquely written as N = a·2^k − 2, with a odd and k ≥ 3. This defines a family of unresolved values. 5. We define a worst-case envelope for the growth phase: Na,n = 2(a·3^n − 1), where n is the number of odd steps. This envelope bounds the maximal value reachable from any a·2^k − 2. 6. We prove that all such envelopes satisfy Na,n ≡ 0 or 4 (mod 8). The 8m case descends immediately to one quarter of its value. Thus, only the 8m+4 envelopes remain unresolved. 7. We formulate the Main Hypothesis (8m+4 Descent Hypothesis): For all odd a and n ≥ 2, if Na,n ≡ 4 mod 8, then the Collatz orbit of Na,n contains a value ≤ (1/4)·Na,n. By Lemma 1, this implies descent to the next level, and iterated application yields global descent. 8. Conclusion: This reduction isolates the essential difficulty of the Collatz problem and provides a finite, testable hypothesis. The contribution is not a proof, but a structural decomposition that identifies the precise core of the problem. 日本語による補足： 本研究は、Collatz問題を「8m+4型包絡線の降下仮説」へと還元する構造的手法を提示する。OK条件による偶数の分類、最悪成長包絡線の導入、合同構造の解析を通じて、問題の未解決部分を有限な仮説に集約する。これは、Collatz問題の本質的困難を明示的に切り出す試みである。