# Archimedean property in real closed fields - where's the mistake?

Let $$\left$$ 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?