Exploring the inverse mapping from a dynamics to a residue class - inputting a reduced dynamics or partial dynamics and outputting a residue class

We study a reverse problem - given a reduced dynamics or partial dynamics, can we compute a residue class who presents that dynamics.

We design a dedicated algorithm that takes as input a dynamics with length t consists of “I” or “O” can output a residue class who present this dynamics in the first t transformations. We thus design computer program that can output a reside class by inputting a reduced dynamics or partial dynamics. That is, inputting c∈{I,O}^L,  CntO(s) ≤ ceil( CntI(s)*lambda), lambda=ln1.5/ln2=0.58469250, s = Substr(c,1,i), i=1,2,..., L. In other words, the dynamics is above of or cutting ratio line in our proposed Collatz graph.

Note that, the algorithm is quit lightweight and designed from our formal proof of Partition Theorem - We prove that all natural numbers are partitioned regularly corresponding to ongoing dynamics. Given any natural number x that is i module 2^t (i is an odd integer), the first t transformations in terms of I or O can be determined and identical with the first t transformations of of i. Once current value after t (t is greater or equal to 2) transformations of I or O, is less than x, then reduced dynamics of x is obtained. Otherwise, the residue class of x (namely, i module 2^t) can be partitioned into two halves (namely, i module 2^{t+1} and i+2^t module 2^{t+1}), and either half presents I or O in intermediately forthcoming (t+1)-th transformation.