Let $\left<F,+,\cdot,\leq\right>$ be a real closed field. Consider the following schema:
$(*)$ Let $\phi(\cdot)$ be a first-order predicate. Let there exist $x \neq y$ such that $\phi(x),\phi(y)$. For all $z$, let $\phi(z)$ imply $\phi(2z-x)$ and $\phi(2x-z)$. Then, for all $t$, there are $x',y':\phi(x'),\phi(y'),x'\leq t\leq y'$.
This schema seems to be a first-order one. This also seems to imply Archimedean property:
We have $(2z-x)-x=2(z-x)$ and $(2x-z)-x=(x-z)$. So if $\phi(y)$ and $|x-z|=2^n|x-y|, n \in \mathbb N$, then $\phi(z)$.
Take any $x', y': x' \leq y'$. By $(*)$, we can find $u,z,t,w: u \leq x' \leq z, t \leq y' \leq w, |x-u|=2^n|x-y|,|x-z|=2^m|x-y|$,$|x-t|=2^k|x-y|,|x-w|=2^l|x-y|$. $u \leq x' \leq y' \leq w$. $|x'-y'| \leq |u - w| \leq (2^l+2^n)|x-y|$.
But I know that, at least under GCH, $\mathbb R$ is the only real closed Archimedean field; I also know that real closed fields are indistinguishable by first order properties.
So, at least one of my assertions has to be false. Which one?