This paper presents an algorithmic method that, given a positive integer $j$, generates the $j$-th convergence stair containing all natural numbers from where the Collatz conjecture holds by exactly $j$ applications of the Collatz function. To this end, we present a novel formulation of the Collatz conjecture as a concurrent program, and provide the general case specification of the $j$-th convergence stair for any $j > 0$. The proposed specifications provide a layered and linearized orientation of Collatz numbers organized in an infinite set of infinite binary trees. To the best of our knowledge, this is the first time that such a general specification is provided, which can have significant applications in analyzing and testing the behaviors of complex non-linear systems. We have implemented this method as a software tool that generates the Collatz numbers of individual stairs. We also show that starting from any value in any convergence stair the conjecture holds. However, to prove the conjecture, one has to show that every natural number will appear in some stair; i.e., the union of all stairs is equal to the set of natural numbers, which remains an open problem.

中文翻译：

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指定和验证 Collat​​z 程序的收敛阶梯

本文提出了一种算法方法，给定正整数 $j$，生成第 $j$ 个收敛阶梯，其中包含 Collat​​z 猜想通过 Collat​​z 函数的 $j$ 应用而成立的所有自然数。为此，我们提出了 Collat​​z 猜想作为并发程序的新颖表述，并为任何 $j > 0$ 提供了第 $j$ 个收敛阶梯的一般情况规范。所提出的规范提供了以无限二叉树的无限集合组织的 Collat​​z 数的分层和线性化方向。据我们所知，这是第一次提供这样的通用规范，它在分析和测试复杂非线性系统的行为方面具有重要的应用。我们已将此方法实现为生成各个楼梯的 Collat​​z 数的软件工具。我们还表明，从任何收敛阶梯的任何值开始，该猜想都成立。然而，为了证明这个猜想，我们必须证明每个自然数都会出现在某个楼梯上；即，所有楼梯的并集等于自然数集，这仍然是一个悬而未决的问题。