= McCleary trichotomy theorem
{c}
A faithful transitive <order-primitive lattice permutation group> is either a regular Archimedean translation action, an order two-transitive action, or a periodic action. In the periodic case a fixed-point-free order automorphism of the <Dedekind completion> commutes with the action and has cofinal and coinitial iterates; the point stabilizers act order two-transitively within period intervals. Only the regular case satisfies $|f||g|\le|g|^2|f|^2$ for every pair. In either other case, order two-transitivity on the whole chain or on a period interval supplies positive $a,b$ and $x<y<z<t$ with $xa=y$, $ya=z$, $xb=x$, $yb=t$, giving $xab=t>z=xb^2a^2$. The classification itself is a theorem, not a consequence of that witness calculation.
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