Status of the Nichols claim to proof of the Collatz conjecture?

Question

In December 2021, Robert Hills Nichols, Jr of Cumberland University claimed a proof of Collatz on arXiv. But I've not seen any reviews or comments on the validity of his claim, and it's certainly beyond me.

Is there any validity to his claim?

https://arxiv.org/abs/2112.07361

Answer 1

Got a quick look and from what I see, there is a bunch of variable transformation and 2 main functions $p_n=p(2^{n+1}x+2^n-1)=x$ which gives the root $x$ of a number of the form $2^{n+1}x+2^n-1$ and some kind of 2-valuation $q_n=q(y)=\nu_2(y+1)+1$ used like this: $q(2^{n+1}x+2^n-1)=n+1$ which leads to the construction of two sequences: $v_n$ which are odd numbers and are the "bottom" of successive 1-cycles (the classical $a2^i-1$), and $u_n$ which are even and are the top of the successive 1-cycles (the classical $a3^i-1$)

e.g. from $v_0=7$, you get $u_1=26$, $v_1=13$, $u_2=20$, $v_2=5$,....

This re-arrangement gives another (Collatz) tree which has the same pitfalls as the classical (Collatz) tree: It fails to show the presence of all natural numbers in the tree (it uses the same argument as the classical tree), it fails to show there are no cycles (several trees), it fails to show there are no infinite sequences.