  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.
