an interesting book Old and new unsolved problems in plane geometry and number theory by Klee/Wagon (1991) includes the Collatz conjecture. on p225 they consider iterates $\mod 2^n$ and state that unless $n$ is congruent modulo 256 to 27, 31, 47, 63, 71, 91, 103, 111, 127, 155, 159, 167, 191, 207, 223, 231, 239, 251, 255 then the stopping time is less than 5 ("exercise 4"). (this seems to be a typo in that exercise 4 is not related to that fact. there is no further analysis other than the simple factoid.)
they point out this approach is useful as a sort of sieving approach to eliminate terminating trajectories and works for arbitrary values $\mod 2^n$.
- looking for more detail/explanation on how this approach works.
- is there a ref that analyzes collatz from this approach?
Hmm, I don't know whether I got the concept of the mod 2^n- idea in your question correctly. Perhaps this is related/interesting in some way.
The following table tabulates all odd numbers $a$ and the odd result of one transformation, calling it $b$. One transformation is here defined as $$ b = {3a+1\over2^A}$$ where $A$ is that exponent which makes the result $b$ an odd number. This $A$ equals also the number $A$ of "even"/"descending" steps in the usual Collatz-notation.
That transformations are equal in some modular classes, and $k \ge 0$ is there a free parameter $$ \begin{array} {r|rrl|rrl} c& a&&\to b &a & &\to b \\ \hline \\ 1& 4^1 k &+3 & \to 6k+5 & 2 \cdot 4^1 k &+1 & \to 6k+1 \\ 2& 4^2 k &+13 & \to 6k+5 & 2 \cdot 4^2 k &+5 & \to 6k+1 \\ 3& 4^3 k &+53 & \to 6k+5 & 2 \cdot 4^3 k &+21 & \to 6k+1 \\ & \vdots \\ c& 4^c k & + \small {10 \cdot 4^{c-1}-1 \over 3} & \to 6k+5 & 2 \cdot 4^c k &+ \small {4^c-1 \over 3} & \to 6k+1 \\ \end{array} $$
Now this is related to the $\pmod {2^n}$- concept and we see, that always the result of one step is smaller than the initial value $a$ except if $a$ is of the form $a = 4k+3$ where $b \gt a$ and if a is of the form $8k+1$ and $k=0$ then $b=a$.
The exponent $c$ gives the number of "immediate" steps / directly decreasing and is the stopping time for all numbers $a$ except from the first entry in the table.
I think, the proposal/observation in the article can then be reproduced if we use this table and apply it for two or even more transformations (I'm not going to do this here but perhaps it is a helpful/interesting view into the modular whereabouts).