Third terms of a fully open path: ν₂(t_j − t_i) is the minimal level between them

A tree witness $\theta:\mathbb N\times\mathbb Z\to\{0,1\}$ defines a strict total order $\prec_\theta$ on $\mathbb Z$ (milestone framework). A finite word $T$ of integers is $\theta$-safe if the sequence "letters of $T$ in order, then all remaining integers in $\prec_\theta$-order" has no monotone 4-term arithmetic progression; positions are first occurrences. For an in-order pair $(a,b)$ of a word ($a$ before $b$), call $t=2b-a$ its third term, $w=3b-2a$ its fourth term, and $\nu_2(b-a)$ its level. An open path in $T$ is a family of in-order pairs $(a_i,b_i)$, $0\le i<m$, of letters of $T$ with $3b_i-2a_i=2b_{i+1}-a_{i+1}$ for $0\le i<m-1$ and no third term $t_i=2b_i-a_i$ in $T$. Write $d_i=b_i-a_i$, $k_i=\nu_2(d_i)$, and $w=3b_{m-1}-2a_{m-1}$ for the fourth term of the last (sink) pair.

Theorem. Suppose $T$ is $\theta$-safe for some tree witness $\theta$ and contains an open path of length $m$ whose sink fourth term $w$ is also absent from $T$. Then for all $0\le i<j<m$ there is $p$ with $i\le p<j$ and $k_p=\min_{i\le l<j}k_l$ such that

and for every $0\le i<m$ likewise $\nu_2(w-t_i)=\min_{i\le l<m}\nu_2(b_l-a_l)$ (the minimum attained at some $p\in[i,m)$).

Along a fully open path the 2-adic distance between third terms (and from a third term to the sink's fourth term) is exactly the lowest level of the pairs in between. This sharpens the ruler law (milestone #148): $t_j-t_i=\sum_{i\le l<j}d_l$, and the ruler law makes the minimal valuation in every window unique.