Fully open paths are short: an open path of length m needs at least log2(m+1) distinct 2-adic levels

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 $2b_i-a_i$ in $T$.

Theorem. Suppose $T$ is $\theta$-safe for some tree witness $\theta$ and contains an open path of length $m$ whose last fourth term $3b_{m-1}-2a_{m-1}$ is also absent from $T$. Let $r=\#\{\nu_2(b_i-a_i): i<m\}$ be the number of distinct levels on the path. Then

and the minimal level along the path is attained by exactly one pair (if $\nu_2(b_i-a_i)$ is minimal and $\nu_2(b_j-a_j)=\nu_2(b_i-a_i)$, then $j=i$).

So a fully open path of length $m$ uses at least $\log_2(m+1)$ distinct 2-adic levels. This is the quantitative form of the ruler law (milestone #148): the level sequence is a ruler sequence, and a ruler sequence with $r$ distinct values has length at most $2^r-1$ (split at the unique minimum and induct on $r$).