The Erdős Squarefree Conjecture via Cubic Number Fields: A Structural Approach to n!+1

New structural framework for the Erdős conjecture using the cubic field Q(α), α³-α-1=0, discriminant -23. Main results: periodicity of square-divisibility sets, density bounds O(1/p⁴) from cubic arithmetic, classification by splitting type, and effective reducibility of Brocard's problem via Thue-Mahler theory.