I have a question about some (seemingly unimportant) behavior I noticed in Collatz sequences, which I haven't been able to find a general answer to upon rough scan of the literature (please be wary that this is not my field, so there may be some ignorance here).

Let $N, \text{Col}(N), \text{Col}^2(N), \text{Col}^3(N), ...$ denote a Collatz orbit with initial value $N$, where for the purpose of this question we will put $c_0 = N = \text{Col}^0(N)$, $c_1 =\text{Col}(N)$, and $c_k =\text{Col}^k(N)$. Further, lets naturally call a Collatz sub-sequence of form $(c_i, c_{i+1}, ...c_{i+(m-1)})$ a length $m$ Collatz tuple.

My question then follows from the following observation: it would appear that all length $m$ Collatz tuples lie on one of $m$ distinct lines in an $m$ dimensional space, where the pairwise angle between each such line is $\pi/4$. This fact is clear for $m=2$, where all length 2 Collatz tuples lie on the lines $y = 3x+1$ and $y = x/2$ which have a intersection angle of $\pi/4$.

That this necessarily holds for the general $m$ case however is not immediately obvious to me (where we take Collatz orbits of length greater than or equal to $m$). I have been able to come up with a variety of arguments for why this may be the case, but none have been entirely convincing or neat. Can someone elucidate why this appears to be so, and if this property fails for some $m$, why? Can you provide sketch of proof (unless I am missing something basic)?