A rational counterexample to the reciprocal-logarithmic assertion in Erdős problem 143

We construct a countably infinite set of rational numbers A contained in [2, infinity) such that |ka-b| >= 1 for all distinct a,b in A and all positive integers k, while the sum of 1/(a log a) over A diverges. The proof is a finite extension argument with a fixed positive weight gain at each round. New points are rational dilations of finite primitive sets selected from periodic sets by an elementary Euler-product argument. Every constructed point lies within m^(-12)/2 to the right of an integer m. A companion Lean 4 project proves the existence theorem and the periodic-selection lemma. The revised logarithmic analytic argument is not formalized line by line. Research manuscript revised October 25, 2026, with standalone LaTeX source and a companion Lean 4 source archive. The archive includes reproduction instructions and dated verification records. This is a preprint.