AIM Problem 1.18 asks a structural question about popular differences. If every represented difference of a finite set occurs more than one third as often as the size of that set, must the whole difference set be a subgroup or a union of at most three cosets? This anonymous, unrefereed candidate gives a negative answer. In the odd cyclic group modulo twenty-one, take the eight elements zero, one, two, seven, nine, fourteen, fifteen, and sixteen. Their difference set has fifteen elements. Every represented difference occurs at least three times, which is strictly more than eight thirds. Yet fifteen is neither a subgroup size of a group of order twenty-one nor the size of at most three cosets of any common subgroup. The same idea has a transparent three-layer form. Two full layers surround one partial layer. Exact counting gives five quotient-difference layers and minimum multiplicity equal to the smaller of the full-layer size and twice the middle-layer size. Choosing the middle layer just above two fifths produces odd cyclic counterexamples for every fixed threshold below two fifths. This is a lower obstruction, not a proof that two fifths is sufficient or sharp. Python and JavaScript checks, mutation tests, fresh-extraction replay, and internal role-separated review pass. They establish producer-side replay, not independent reconstruction, formal verification, specialist review, journal peer review, novelty, or priority. The scholarly creator is Anonymous. This synthetic-voice briefing is a communication aid, not additional mathematical evidence.