The isolated-minimum completion has no isolated-maximum analogue

A tree witness is a function $\theta(k,r)\in\{0,1\}$ on the nodes $(k,r)$, $0\le r<2^k$, of the binary tree of residues. For distinct integers $x,y$ with $k=\nu_2(y-x)$, put $x\prec_\theta y$ iff $\mathrm{bit}_k(x)=\theta(k,\,x\bmod 2^k)$ (two's-complement digits). A finite word $Q$ of integers is $\theta$-safe if the sequence "the letters of $Q$ in order, then all remaining integers in $\prec_\theta$-order" contains no monotone $4$-term arithmetic progression; it is tree-safe if it is $\theta$-safe for some $\theta$. The isolated-minimum completion (if $b$ is the $\theta$-minimum of a $\theta$-safe word $P$, alone in its class mod $2^{\nu_2(b-a)+1}$, and $3b-2a$ is not a third term $2v-u$ of an in-order pair, then $P\cdot(3b-2a)$ is tree-safe) has no maximum analogue: there exist $\theta$, $P$, $a\neq b$ with $P$ $\theta$-safe, $b\in P$ the $\theta$-maximum of $P$, $b$ alone in its class mod $2^{\nu_2(b-a)+1}$, and $3b-2a\neq 2v-u$ for all in-order pairs $(u,v)$ of $P$, such that

A witness is $\theta\equiv1$, $P=(0,1)$, $a=1$, $b=0$: the pairs $(0,1)$ and $(1,-2)$ force opposite values at the root (Erdős Problem 195).

Formalization note: treeLt θ is from Definitions.Def_Erdos195_tree; Safe, bitOne are from Definitions.Def_Erdos195_defs. Positions in a word are List.idxOf; P ++ [c] appends c.