Lacunary Counterexamples to a Distinct-Summand Freiman Container Problem

Fix K > 1. We construct finite nonempty sets A,B ⊆ ℤ with |A| > |B| and |A+B| < K|A| for which B is not contained in any generalized arithmetic progression of bounded rank and size O_K(|A|). More strongly, for every prescribed rank bound d and size constant C, one counterexample defeats all progressions of rank at most d and cardinality at most C|A|. The construction is B = {0,1,N,…,Nʳ⁻¹} and A = kB. Uniqueness of base-N digits gives exact polynomial growth |sB| = (s+r choose r) for s < N, whereas every rank-d generalized arithmetic progression has h-fold growth at most h^d. This answers Problem 2.5 from the 2004 AIM list on recent trends in additive combinatorics in the negative for every K > 1. Research status: Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-use disclosure: AI-assisted tools supported literature search, computation, proof auditing, and manuscript preparation. The author remains responsible for all claims and the final text.